@inproceedings{avw-isdcm-23,
author = {Oswin Aichholzer and Birgit Vogtenhuber and Alexandra Weinberger},
title = {{{Isomorphisms of simple drawings of complete multipartite graphs}}},
booktitle = {Abstracts of the XX Spanish Meeting on Computational Geometry},
pages = {59},
year = 2023,
address = {Santiago de Compostela, Spain},
url = {https://egc23.web.uah.es/wp-content/uploads/2023/07/EGC2023_Booklet.pdf#page=71},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{agtvw-rrsgt-23,
author = {Oswin Aichholzer and Alfredo Garc{\'i}a and Javier Tejel and Birgit Vogtenhuber and Alexandra Weinberger},
title = {{{Recognizing rotation systems of generalized twisted drawings in $O(n^2)$ time}}},
booktitle = {Abstracts of the XX Spanish Meeting on Computational Geometry},
pages = {69},
year = 2023,
address = {Santiago de Compostela, Spain},
url = {https://egc23.web.uah.es/wp-content/uploads/2023/07/EGC2023_Booklet.pdf#page=81},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{agtvw-crsgt-23,
author = {Oswin Aichholzer and Alfredo Garc{\'i}a and Javier Tejel and Birgit Vogtenhuber and Alexandra Weinberger},
title = {{{Characterizing rotation systems of generalized twisted drawings via 5-tuples}}},
booktitle = {Abstracts of the XX Spanish Meeting on Computational Geometry},
pages = {71},
year = 2023,
address = {Santiago de Compostela, Spain},
url = {https://egc23.web.uah.es/wp-content/uploads/2023/07/EGC2023_Booklet.pdf#page=83},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ahvv-nng4h-16,
author = {Oswin Aichholzer and Thomas Hackl and Pavel Valtr and Birgit Vogtenhuber},
title = {{{A Note on the Number of General 4-holes in (Perturbed) Grids}}},
booktitle = {Discrete and Computational Geometry and Graphs. JCDCGG 2015.},
series = {Lecture Notes in Computer Science (LNCS)},
volume = {9943},
editor = {Akiyama, Jin and Ito, Hiro and Sakai, Toshinori and Uno, Yushi},
publisher = {Springer, Cham},
pages = {1--12},
year = 2016,
e-isbn = {978-3-319-48532-4"},
isbn = {978-3-319-48531-7},
doi = {https://doi.org/10.1007/978-3-319-48532-4_1},
abstract = {{Considering a variation of the classical Erd{\H{o}}s-Szekeres type
problems, we count the number of general 4-holes (not necessarily convex, empty 4-gons)
in squared Horton sets of size $\sqrt{n}\!\times\!\sqrt{n}$.
Improving on previous upper and lower bounds we
show that this number is $\Theta(n^2\log n)$, which constitutes
the currently best upper bound on minimizing the number of general
\mbox{$4$-holes} for any set of $n$ points in the plane.
To obtain the improved bounds, we prove a result of independent
interest. We show that $\sum_{d=1}^n \frac{\varphi(d)}{d^2} =
\Theta(\log n)$, where $\varphi(d)$ is Euler's phi-function, the
number of positive integers less than~$d$ which are relatively prime
to $d$. This arithmetic function is also called Euler's totient
function and plays a role in number theory and cryptography.}},
originalfile = {/geometry/cggg.bib}
}
@article{afhpruv-clcfc-18,
author = {Oswin Aichholzer and Ruy Fabila-Monroy and Ferran Hurtado
and Pablo Perez-Lantero and Andres J. Ruiz-Vargas and Jorge Urrutia and Birgit Vogtenhuber},
title = {{{Cross-sections of line configurations in {$R^3$} and $(d\!-\!2)$-flat configurations in {$R^d$}}}},
journal = {Computational Geometry: Theory and Applications},
volume = {77},
number = {},
pages = {51--61},
year = {2019},
note = {Special Issue of CCCG 2014},
issn = {0925-7721},
doi = {https://doi.org/10.1016/j.comgeo.2018.02.005},
url = {http://www.sciencedirect.com/science/article/pii/S0925772118300154},
abstract = {{We consider sets $\mathcal{L} = \{\ell_1, \ldots, \ell_n\}$ of $n$ labeled lines in general position in ${{\sf l} \kern -.10em {\sf R} }^3$, and
study the order types of point sets $\{p_1, \ldots, p_n\}$ that stem from the intersections of the lines in $\mathcal{L}$ with (directed) planes $\Pi$, not parallel to any line of $\mathcal{L}$, that is, the proper cross-sections of $\mathcal{L}$.
As two main results, we show that the number of different order types that can be obtained as cross-sections of $\mathcal{L}$ is $O(n^9)$ when considering all possible planes $\Pi$,
and $O(n^3)$ when restricting considerations to sets of pairwise parallel planes, where both bounds are tight.
The result for parallel planes implies that any set of $n$ points in ${{\sf l} \kern -.10em {\sf R} }^2$ moving with constant (but possibly different) speeds along straight lines forms at most $O(n^3)$ different order types over time.
We further generalize the setting from ${{\sf l} \kern -.10em {\sf R} }^3$ to ${{\sf l} \kern -.10em {\sf R} }^d$ with $d>3$, showing that the number of order types that can be obtained as cross-sections of a set of $n$ labeled $(d\!-\!2)$-flats in ${{\sf l} \kern -.10em {\sf R} }^d$ with planes is
$O\left(\dbinom{\binom{n}{3}+n}{d(d\!-\!2)}\right)$.}},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ahkprrrv-psst-16,
author = {Oswin Aichholzer and Thomas Hackl and Matias Korman and Alexander Pilz and G{\"u}nter Rote and Andr{\'e} van Renssen and Marcel Roeloffzen and Birgit Vogtenhuber},
title = {{{Packing Short Plane Spanning Trees in Complete Geometric Graphs}}},
booktitle = {27th International Symposium on Algorithms and Computation (ISAAC 2016)},
pages = {9:1--9:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
isbn = {978-3-95977-026-2},
issn = {1868-8969},
year = {2016},
volume = {64},
editor = {Seok-Hee Hong},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
url = {http://drops.dagstuhl.de/opus/volltexte/2016/6782},
urn = {urn:nbn:de:0030-drops-67823},
doi = {10.4230/LIPIcs.ISAAC.2016.9},
annote = {Keywords: Geometric Graphs, Graph Packing, Plane Graphs, Minimum Spanning Tree, Bottleneck Edge},
abstract = {Given a set of points in the plane, we want to establish a connection network between these points that consists of several disjoint layers. Motivated by sensor networks, we want that each layer is spanning and plane, and that no edge is very long (when compared to the minimum length needed to obtain a spanning graph). We consider two different approaches: first we show an almost optimal centralized approach to extract two graphs. Then we show a constant factor approximation for a distributed model in which each point can compute its adjacencies using only local information. In both cases the obtained layers are plane.},
originalfile = {/geometry/cggg.bib}
}
@article{aaghlru-gapi-13,
author = {O. Aichholzer and G.~Araujo-Pardo and N.~Garc{\'i}a-Col{\'i}n and T.~Hackl and D.~Lara and C.~Rubio-Montiel and J.~Urrutia},
title = {{{Geometric achromatic and pseudoachromatic indices}}},
journal = {Graphs and Combinatorics},
year = 2016,
category = {3a},
volume = {32},
number = {2},
pages = {431--451},
thackl_label = {50J},
htmlnote = {
Springer Online First.},
pdf = {/files/publications/geometry/aaghlru-gapi-13.pdf},
doi = {http://dx.doi.org/10.1007/s00373-015-1610-x},
arxiv = {1303.4673},
abstract = {The pseudoachromatic index of a graph is the
maximum number of colors that can be assigned to its
edges, such that each pair of different colors is
incident to a common vertex. If for each vertex its
incident edges have different color, then this
maximum is known as achromatic index. Both indices
have been widely studied. A geometric graph is a
graph drawn in the plane such that its vertices are
points in general position, and its edges are
straight-line segments. In this paper we extend the
notion of pseudoachromatic and achromatic indices
for geometric graphs, and present results for
complete geometric graphs. In particular, we show
that for n points in convex position the achromatic
index and the pseudoachromatic index of the complete
geometric graph are
$\lfloor\frac{n^2+n}{4}\rfloor$.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ahvv-nng4h-15,
author = {O. Aichholzer and T.~Hackl and P.~Valtr and
B.~Vogtenhuber},
title = {{{A note on the number of general 4-holes in
perturbed grids}}},
booktitle = {Proc. $18^{th}$ Japan Conference on Discrete and
Computational Geometry and Graphs (JCDCG$^2$ 2015)},
pages = {68--69},
year = 2015,
address = {Kyoto, Japan},
category = {3b},
thackl_label = {49C},
xxarxiv = {},
xpdf = {/files/publications/geometry/ahvv-nng4h-15.pdf},
abstract = {Considering a variation of the classical Erd{\H{o}}s
and Szekeres type problems, we count the number of
general $4$-holes (empty \mbox{$4$-gons}) in the
$\sqrt{n}\!\times\!\sqrt{n}$ squared Horton
set. Improving on previous upper and lower bounds we
show that this number is $\Theta(n^2\log n)$, which
also constitutes the currently best upper bound on
minimizing the number of general \mbox{$4$-holes}
for any set of $n$ points in the plane.\\ To obtain
these bounds and as a result of independent
interest, we show that $\sum_{d=1}^n
\frac{\varphi(d)}{d^2} = \Theta(\log n)$, where
$\varphi(d)$ is Euler's phi-function, the number of
positive integers less than~$d$ which are relatively
prime to $d$.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{akmpw-oarps-15,
author = {O. Aichholzer and V. Kusters and W. Mulzer and A. Pilz and M. Wettstein},
title = {{{An optimal algorithm for reconstructing point set order types from radial orderings}}},
booktitle = {Proceedings $26^{th}$ Int. Symp. Algorithms and Computation (ISAAC 2015)},
pages = {505--516},
eprint = {1507.08080},
archiveprefix = {arXiv},
year = 2015,
category = {3b},
abstract = {Given a set $P$ of $n$ labeled points in the plane, the radial system of~$P$ describes, for each $p\in P$, the radial order of the other points around~$p$.
This notion is related to the order type of~$P$, which describes the orientation (clockwise or counterclockwise) of every ordered triple of~$P$.
Given only the order type of $P$, it is easy to reconstruct the radial system of $P$, but the converse is not true.
Aichholzer et~al.\ (Reconstructing Point Set Order Types from Radial Orderings, in Proc.~ISAAC 2014) defined $T(R)$ to be the set of order types with radial system~$R$ and showed that sometimes $|T(R)|=n-1$.
They give polynomial-time algorithms to compute $T(R)$ when only given~$R$.
We describe an optimal $O(kn^2)$ time algorithm for computing $T(R)$, where $k$ is the number of order types reported by the algorithm.
The reporting relies on constructing the convex hulls of all possible point sets with the given radial system, after which sidedness queries on point triples can be answered in constant time.
This set of convex hulls can be constructed in linear time.
Our results generalize to abstract order types.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{abhhhpv-rdtss-15,
author = {O. Aichholzer and T.~Biedl and T.~Hackl
and M.~Held and S.~Huber and P.~Palfrader and B.~Vogtenhuber},
title = {{{Representing Directed Trees as Straight Skeletons}}},
booktitle = {Proc. $23^{nd}$ International Symposium on Graph Drawing (GD 2015)},
series = {Lecture Notes in Computer Science (LNCS)},
volume = {9411},
nopublisher = {Springer, Heidelberg},
editor = {Emilio Di Giacomo and Anna Lubiw},
pages = {335--347},
year = 2015,
address = {Los Angeles, CA, USA},
category = {3b},
thackl_label = {48C},
arxiv = {1508.01076},
doi = {http://dx.doi.org/10.1007/978-3-319-27261-0},
isbn = {978-3-319-27260-3},
e-isbn = {978-3-319-27261-0},
issn = {0302-9743},
pdf = {/files/publications/geometry/abhhhpv-rdtss-15.pdf},
abstract = {The straight skeleton of a polygon is the geometric
graph obtained by tracing the vertices during a
mitered offsetting process. It is known that the
straight skeleton of a simple polygon is a tree, and
one can naturally derive directions on the edges of
the tree from the propagation of the shrinking
process.\\ In this paper, we ask the reverse
question: Given a tree with directed edges, can it
be the straight skeleton of a polygon? And if so,
can we find a suitable simple polygon? We answer
these questions for all directed trees where the
order of edges around each node is fixed.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aafhpprsv-agdsc-15,
author = {B.M.~\'Abrego and O. Aichholzer and S.~Fern\'andez-Merchant and T.~Hackl
and J.~Pammer and A.~Pilz and P.~Ramos and G.~Salazar and B.~Vogtenhuber},
title = {{{All Good Drawings of Small Complete Graphs}}},
booktitle = {Proc. $31^{st}$ European Workshop on Computational
Geometry EuroCG '15},
pages = {57--60},
year = 2015,
address = {Ljubljana, Slovenia},
category = {3b},
thackl_label = {46C},
xxarxiv = {},
pdf = {/files/publications/geometry/aafhpprsv-agdsc-15.pdf},
abstract = {\emph{Good drawings} (also known as \emph{simple
topological graphs}) are drawings of graphs such
that any two edges intersect at most once. Such
drawings have attracted attention as generalizations
of geometric graphs, in connection with the crossing
number, and as data structures in their own right.
We are in particular interested in good drawings of
the complete graph. In this extended abstract, we
describe our techniques for generating all different
weak isomorphism classes of good drawings of the
complete graph for up to nine vertices. In
addition, all isomorphism classes were enumerated.
As an application of the obtained data, we present
several existential and extremal properties of these
drawings.},
originalfile = {/geometry/cggg.bib}
}
@article{aahhpruvv-kcps-14,
author = {O. Aichholzer and F.~Aurenhammer and T.~Hackl and
F.~Hurtado and A.~Pilz and P.~Ramos and J.~Urrutia and P.~Valtr and B.~Vogtenhuber},
title = {{{On $k$-Convex Point Sets}}},
journal = {Computational Geometry: Theory and Applications},
year = 2014,
volume = {47},
number = {8},
pages = {809--832},
thackl_label = {40J},
category = {3a},
issn = {0925-7721},
url = {http://www.sciencedirect.com/science/article/pii/S0925772114000534},
doi = {http://dx.doi.org/10.1016/j.comgeo.2014.04.004},
pdf = {/files/publications/geometry/aahhpruvv-kcps-14.pdf},
abstract = {We extend the (recently introduced) notion of
$k$-convexity of a two-di\-men\-sional subset of the
Euclidean plane to finite point sets. A set of $n$
points is considered $k$-convex if there exists a
spanning (simple) polygonization such that the
intersection of any straight line with its interior
consists of at most~$k$ disjoint intervals. As the
main combinatorial result, we show that every
$n$-point set contains a subset of $\Omega(\log^2
n)$ points that are in 2-convex position. This
bound is asymptotically tight. From an algorithmic
point of view, we show that 2-convexity of a finite
point set can be decided in polynomial time, whereas
the corresponding problem on $k$-convexity becomes
NP-complete for any fixed $k\geq 3$.},
originalfile = {/geometry/cggg.bib}
}
@article{ahplmv-mseup-15,
author = {O. Aichholzer and T.~Hackl
and S.~Lutteropp and T.~Mchedlidze and A.~Pilz and B.~Vogtenhuber},
title = {{{Monotone Simultaneous Embedding of Upward Planar Digraphs}}},
journal = {Journal of Graph Algorithms and Applications},
year = 2015,
volume = {19},
number = {1},
pages = {87--110},
thackl_label = {39J},
category = {3a},
issn = {1526-1719},
doi = {http://dx.doi.org/10.7155/jgaa.00350},
arxiv = {1310.6955v2},
pdf = {/files/publications/geometry/ahplmv-mseup-15.pdf},
abstract = {We study monotone simultaneous embeddings of upward
planar digraphs, which are simultaneous embeddings
where the drawing of each digraph is upward planar,
and the directions of the upwardness of different
graphs can differ. We first consider the special
case where each digraph is a directed path. In
contrast to the known result that any two directed
paths admit a monotone simultaneous embedding, there
exist examples of three paths that do not admit such
an embedding for any possible choice of directions
of monotonicity.\\ We prove that if a monotone
simultaneous embedding of three paths exists then it
also exists for any possible choice of directions of
monotonicity. We provide a polynomial-time
algorithm that, given three paths, decides whether a
monotone simultaneous embedding exists and, in the
case of existence, also constructs such an
embedding. On the other hand, we show that already
for three paths, any monotone simultaneous embedding
might need a grid whose size is exponential in the
number of vertices. For more than three paths, we
present a polynomial-time algorithm that, given any
number of paths and predefined directions of
monotonicity, decides whether the paths admit a
monotone simultaneous embedding with respect to the
given directions, including the construction of a
solution if it exists. Further, we show several
implications of our results on monotone simultaneous
embeddings of general upward planar digraphs.
Finally, we discuss complexity issues related to our
problems.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ahplmv-msedp-14,
author = {O. Aichholzer and T.~Hackl
and S.~Lutteropp and T.~Mchedlidze and A.~Pilz and B.~Vogtenhuber},
title = {{{Monotone Simultaneous Embedding of Directed Paths}}},
booktitle = {Proc. $30^{th}$ European Workshop on Computational
Geometry EuroCG '14},
pages = {online},
year = 2014,
address = {Dead Sea, Israel},
category = {3b},
thackl_label = {39C},
arxiv = {1310.6955v2},
pdf = {/files/publications/geometry/ahplmv-msedp-14.pdf},
abstract = {We consider a variant of monotone simultaneous
embeddings (MSEs) of directed graphs where all
graphs are directed paths and have distinct
directions of monotonicity. In contrast to the
known result that any two directed paths admit an
MSE, there exist examples of three paths that do not
admit such an embedding for any possible choice of
directions of monotonicity. We prove that if an MSE
of three paths exists then it also exists for any
possible choice of directions of monotonicity. We
provide a polynomial-time algorithm that, given
three paths, decides whether an MSE exists.
Finally, we provide a polynomial-time algorithm that
answers the existence question for any given number
of paths and predefined directions of monotonicity.},
originalfile = {/geometry/cggg.bib}
}
@article{ahkpv-gpps-14,
author = {O. Aichholzer and T.~Hackl and M.~Korman and A.~Pilz and
B.~Vogtenhuber},
title = {{{Geodesic-preserving polygon simplification}}},
journal = {Int'l Journal of Computational Geometry \& Applications},
year = 2014,
volume = {24},
category = {3a},
number = {4},
oaich_label = {},
thackl_label = {38J},
pages = {307--323},
pdf = {/files/publications/geometry/ahkpv-gpps-13.pdf},
doi = {http://dx.doi.org/10.1142/S0218195914600097},
arxiv = {1309.3858},
abstract = {Polygons are a paramount data structure in computational
geometry. While the complexity of many algorithms on simple
polygons or polygons with holes depends on the size of the
input polygon, the intrinsic complexity of the problems
these algorithms solve is often related to the reflex
vertices of the polygon. In this paper, we give an
easy-to-describe linear-time method to replace an input
polygon~$P$ by a polygon $P'$ such that (1)~$P'$
contains~$P$, (2)~$P'$ has its reflex vertices at the same
positions as~$P$, and (3)~the number of vertices of $P'$ is
linear in the number of reflex vertices. Since the
solutions of numerous problems on polygons (including
shortest paths, geodesic hulls, separating point sets, and
Voronoi diagrams) are equivalent for both $P$ and $P'$, our
algorithm can be used as a preprocessing step for several
algorithms and makes their running time dependent on the
number of reflex vertices rather than on the size of~$P$.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ahkpv-gpps-13,
author = {O. Aichholzer and T.~Hackl and M.~Korman and A.~Pilz and
B.~Vogtenhuber},
title = {{{Geodesic-preserving polygon simplification}}},
booktitle = {Lecture Notes in Computer Science (LNCS), Proc. $24^{th}$ Int. Symp.
Algorithms and Computation (ISAAC 2013)},
pages = {11--21},
year = {2013},
volume = {8283},
address = {Hong Kong, China},
publisher = {Springer Verlag},
category = {3b},
pdf = {/files/publications/geometry/ahkpv-gpps-13.pdf},
thackl_label = {38C},
arxiv = {1309.3858},
abstract = {Polygons are a paramount data structure in computational
geometry. While the complexity of many algorithms on simple
polygons or polygons with holes depends on the size of the
input polygon, the intrinsic complexity of the problems
these algorithms solve is often related to the reflex
vertices of the polygon. In this paper, we give an
easy-to-describe linear-time method to replace an input
polygon~$P$ by a polygon $P'$ such that (1)~$P'$
contains~$P$, (2)~$P'$ has its reflex vertices at the same
positions as~$P$, and (3)~the number of vertices of $P'$ is
linear in the number of reflex vertices. Since the
solutions of numerous problems on polygons (including
shortest paths, geodesic hulls, separating point sets, and
Voronoi diagrams) are equivalent for both $P$ and $P'$, our
algorithm can be used as a preprocessing step for several
algorithms and makes their running time dependent on the
number of reflex vertices rather than on the size of~$P$.},
originalfile = {/geometry/cggg.bib}
}
@article{afhhj-ems-13,
author = {O. Aichholzer and R.~Fabila-Monroy and T.~Hackl and
C.~Huemer and J.~Urrutia},
title = {{{Empty Monochromatic Simplices}}},
journal = {Discrete \& Computational Geometry},
year = 2014,
volume = {51},
category = {3a},
number = {2},
oaich_label = {},
thackl_label = {37J},
pages = {362--393},
pdf = {/files/publications/geometry/afhhu-ems-13.pdf},
doi = {http://dx.doi.org/10.1007/s00454-013-9565-2},
arxiv = {1210.7043},
abstract = {Let $S$ be a $k$-colored (finite) set of $n$ points in
${R}^d$, $d\geq 3$, in general position, that is, no
\mbox{$(d\!+\!1)$} points of $S$ lie in a common
\mbox{$(d\!-\!1)$}-dimensional hyperplane. We count the
number of empty monochromatic $d$-simplices determined by
$S$, that is, simplices which have only points from one
color class of $S$ as vertices and no points of $S$ in
their interior. For $3 \leq k \leq d$ we provide a lower
bound of $\Omega(n^{d-k+1+2^{-d}})$ and strengthen this to
$\Omega(n^{d-2/3})$ for $k=2$. On the way we provide
various results on triangulations of point sets in~${R}^d$.
In particular, for any constant dimension $d\geq3$, we
prove that every set of $n$ points ($n$ sufficiently
large), in general position in ${R}^d$, admits a
triangulation with at least $dn+\Omega(\log n)$ simplices.},
originalfile = {/geometry/cggg.bib}
}
@article{achhkpsuvw-cpmbl-13a,
author = {O. Aichholzer and J.~Cardinal and T.~Hackl and F.~Hurtado
and M.~Korman and A.~Pilz and R.I.~Silveira and R.~Uehara
and B.~Vogtenhuber and E.~Welzl},
title = {{{Cell-Paths in Mono- and Bichromatic Line
Arrangements in the Plane}}},
journal = {Discrete Mathematics \& Theoretical Computer Science
(DMTCS)},
category = {3a},
oaich_label = {},
thackl_label = {35J},
pages = {317--332},
volume = {16},
number = {3},
year = 2014,
pdf = {/files/publications/geometry/achhkpsuvw-cpmbl-13.pdf},
url = {https://www.dmtcs.org/dmtcs-ojs/index.php/dmtcs/article/view/2590.1.html},
xdoi = {http://dx.doi.org/...},
abstract = {We show that in every arrangement of $n$ red and blue
lines (in general position and not all of the same color)
there is a path through a linear number of cells where red
and blue lines are crossed alternatingly (and no cell is
revisited). When all lines have the same color, and hence
the preceding alternating constraint is dropped, we prove
that the dual graph of the arrangement always contains a
path of length $\Theta(n^2)$.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{achhkpsuvw-cpmbl-13,
author = {O. Aichholzer and J.~Cardinal and T.~Hackl and F.~Hurtado
and M.~Korman and A.~Pilz and R.I.~Silveira and R.~Uehara
and B.~Vogtenhuber and E.~Welzl},
title = {{{Cell-Paths in Mono- and Bichromatic Line
Arrangements in the Plane}}},
booktitle = {Proc. $25^{th}$ Annual Canadian Conference on
Computational Geometry CCCG 2013},
pages = {169--174},
year = 2013,
address = {Waterloo, Ontario, Canada},
category = {3b},
thackl_label = {35C},
pdf = {/files/publications/geometry/achhkpsuvw-cpmbl-13.pdf},
abstract = {We show that in every arrangement of $n$ red and blue
lines (in general position and not all of the same color)
there is a path through a linear number of cells where red
and blue lines are crossed alternatingly (and no cell is
revisited). When all lines have the same color, and hence
the preceding alternating constraint is dropped, we prove
that the dual graph of the arrangement always contains a
path of length $\Theta(n^2)$.},
originalfile = {/geometry/cggg.bib}
}
@article{ahprsv-etgdc-15,
author = {O.~Aichholzer and T.~Hackl and A.~Pilz and P.~Ramos and
V.~Sacrist\'{a}n and B.~Vogtenhuber},
title = {{{Empty triangles in good drawings of the complete graph}}},
journal = {Graphs and Combinatorics},
issn = {0911-0119},
publisher = {Springer Japan},
pages = {335--345},
volume = {31},
number = {2},
htmlnote = {For
Springer Online First.},
doi = {http://dx.doi.org/10.1007/s00373-015-1550-5},
year = 2015,
thackl_label = {34J},
category = {3a},
pdf = {/files/publications/geometry/ahprsv-etgdc-13.pdf},
eprint = {1306.5081},
archiveprefix = {arXiv},
keywords = {Good drawings; Empty triangles; Erdős–Szekeres type problems},
abstract = {A good drawing of a simple graph is a drawing on the
sphere or, equivalently, in the plane in which vertices are
drawn as distinct points, edges are drawn as Jordan arcs
connecting their end vertices, and any pair of edges
intersects at most once. In any good drawing, the edges of
three pairwise connected vertices form a Jordan curve which
we call a triangle. We say that a triangle is empty if one
of the two connected components it induces does not contain
any of the remaining vertices of the drawing of the graph.
We show that the number of empty triangles in any good
drawing of the complete graph $K_n$ with $n$ vertices is at
least $n$.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ahprsv-etgdc-13,
author = {O.~Aichholzer and T.~Hackl and A.~Pilz and P.~Ramos and
V.~Sacrist\'{a}n and B.~Vogtenhuber},
title = {{{Empty triangles in good drawings of the complete graph}}},
booktitle = {Mexican Conference on Discrete Mathematics and
Computational Geometry},
pages = {21--29},
year = 2013,
address = {Oaxaca, M{\'e}xico},
thackl_label = {34C},
category = {3b},
pdf = {/files/publications/geometry/ahprsv-etgdc-13.pdf},
eprint = {1306.5081},
abstract = {A good drawing of a simple graph is a drawing on the
sphere or, equivalently, in the plane in which vertices are
drawn as distinct points, edges are drawn as Jordan arcs
connecting their end vertices, and any pair of edges
intersects at most once. In any good drawing, the edges of
three pairwise connected vertices form a Jordan curve which
we call a triangle. We say that a triangle is empty if one
of the two connected components it induces does not contain
any of the remaining vertices of the drawing of the graph.
We show that the number of empty triangles in any good
drawing of the complete graph $K_n$ with $n$ vertices is at
least $n$.},
originalfile = {/geometry/cggg.bib}
}
@article{ahopsv-fcppt-14,
author = {O.~Aichholzer and T.~Hackl and D.~Orden and A.~Pilz and
M.~Saumell and B.~Vogtenhuber},
title = {{{Flips in combinatorial pointed
pseudo-triangulations with face degree at most four}}},
journal = {Int'l Journal of Computational Geometry \& Applications},
year = 2014,
volume = {24},
number = {3},
pages = {197--224},
thackl_label = {33J},
category = {3a},
issn = {0218-1959},
online-issn = {1793-6357},
doi = {http://dx.doi.org/10.1142/S0218195914600036},
eprint = {1310.0833},
archiveprefix = {arXiv},
pdf = {/files/publications/geometry/ahopsv-fcppt-14.pdf},
abstract = {In this paper we consider the flip operation for
combinatorial pointed pseudo-triangulations where faces
have size~3 or~4, so-called \emph{combinatorial 4-PPTs}. We
show that every combinatorial 4-PPT is stretchable to a
geometric pseudo-triangulation, which in general is not the
case if faces may have size larger than 4. Moreover, we
prove that the flip graph of combinatorial 4-PPTs with
triangular outer face is connected and has dia\/meter
$O(n^2)$.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ahopsv-fcppt-13,
author = {O.~Aichholzer and T.~Hackl and D.~Orden and A.~Pilz and
M.~Saumell and B.~Vogtenhuber},
title = {{{Flips in combinatorial pointed
pseudo-triangulations with face degree at most four
(extended abstract)}}},
booktitle = {Proc. $15^{th}$ Spanish Meeting on Computational Geometry
2013},
pages = {131--134},
year = 2013,
address = {Sevilla, Spain},
thackl_label = {33C},
category = {3b},
eprint = {1310.0833},
pdf = {/files/publications/geometry/ahopsv-fcppt-13.pdf},
abstract = {In this paper we consider the flip operation for
combinatorial pointed pseudo-triangulations where faces
have size~3 or~4, so-called \emph{combinatorial 4-PPTs}. We
show that every combinatorial 4-PPT is stretchable to a
geometric pseudo-triangulation, which in general is not the
case if faces may have size larger than 4. Moreover, we
prove that the flip graph of combinatorial 4-PPTs with
triangular outer face is connected and has dia\/meter
$O(n^2)$.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ahsvw-sdala-13,
author = {O.~Aichholzer and T.~Hackl and V.~Sacrist\'{a}n and
B.~Vogtenhuber and R.~Wallner},
title = {{{Simulating distributed algorithms for lattice agents}}},
booktitle = {Proc. $15^{th}$ Spanish Meeting on Computational Geometry
2013},
pages = {81--84},
year = 2013,
address = {Sevilla, Spain},
thackl_label = {32C},
category = {3b},
pdf = {/files/publications/geometry/ahsvw-sdala-13.pdf},
htmlnote = {Link
to simulators page},
abstract = {We present a practical Java tool for simulating
synchronized distributed algorithms on sets of 2- and
3-dimensional square/cubic lattice-based agents. This
\emph{AgentSystem} assumes that each agent is capable to
change position in the lattice and that neighboring agents
can attach and detach from each other. In addition, it
assumes that each module has some constant size memory and
computation capability, and can send/receive constant size
messages to/from its neighbors. The system allows the user
to define sets of agents and sets of rules and apply one to
the other. The \emph{AgentSystem} simulates the
synchronized execution of the set of rules by all the
modules, and can keep track of all actions made by the
modules at each step, supporting consistency warnings and
error checking. Our intention is to provide a useful tool
for the researchers from geometric distributed algorithms.},
originalfile = {/geometry/cggg.bib}
}
@article{afhhpv-lbnsc-14,
author = {O. Aichholzer and R.~Fabila-Monroy and T.~Hackl and
C.~Huemer and A.~Pilz and B.~Vogtenhuber},
title = {{{Lower bounds for the number of small convex $k$-holes}}},
journal = {Computational Geometry: Theory and Applications},
year = 2014,
volume = {47},
number = {5},
pages = {605--613},
thackl_label = {31J},
category = {3a},
doi = {http://dx.doi.org/10.1016/j.comgeo.2013.12.002},
pdf = {/files/publications/geometry/afhhpv-lbnsc-13-cgta.pdf},
abstract = {Let $S$ be a set of $n$ points in the plane in general
position, that is, no three points of $S$ are on a line. We
consider an Erd{\H{o}}s-type question on the least number
$h_k(n)$ of convex \mbox{$k$-holes} in $S$, and give
improved lower bounds on $h_k(n)$, for $3\leq k\leq 5$.
Specifically, we show that $h_{3}(n) \geq n^2 -
\frac{32n}{7} + \frac{22}{7}$, $h_{4}(n) \geq \frac{n^2}{2}
- \frac{9n}{4} - o(n)$, and $h_5(n) \geq \frac{3n}{4} -
o(n)$. We further settle several questions on sets of 12
points posed by Dehnhardt in 1987.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{afhhpv-lbnsc-12,
author = {O. Aichholzer and R.~Fabila-Monroy and T.~Hackl and
C.~Huemer and A.~Pilz and B.~Vogtenhuber},
title = {{{Lower bounds for the number of small convex $k$-holes}}},
booktitle = {Proc. $24^{th}$ Annual Canadian Conference on
Computational Geometry CCCG 2012},
pages = {247--252},
year = 2012,
address = {Charlottetown, PEI, Canada},
category = {3b},
thackl_label = {31C},
pdf = {/files/publications/geometry/afhhpv-lbnsc-12-cccg.pdf},
abstract = {Let $S$ be a set of $n$ points in the plane in general
position, that is, no three points of $S$ are on a line. We
consider an Erd{\H{o}}s-type question on the least number
$h_k(n)$ of convex \mbox{$k$-holes} in $S$, and give
improved lower bounds on $h_k(n)$, for $3\leq k\leq 5$.
Specifically, we show that $h_{3}(n) \geq n^2 -
\frac{32n}{7} + \frac{22}{7}$, $h_{4}(n) \geq \frac{n^2}{2}
- \frac{9n}{4} - o(n)$, and $h_5(n) \geq \frac{3n}{4} - o(n)$.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{acdhhlr-wmtss-12a,
author = {O. Aichholzer and H.~Cheng and S.L.~Devadoss and T.~Hackl
and S.~Huber and B.~Li and A.~Risteski},
title = {{{What makes a Tree a Straight Skeleton?}}},
booktitle = {Proc. $24^{th}$ Annual Canadian Conference on
Computational Geometry CCCG 2012},
pages = {253--258},
xpages = {267--272},
year = 2012,
address = {Charlottetown, PEI, Canada},
category = {3b},
thackl_label = {30C},
pdf = {/files/publications/geometry/acdhhlr-wmtss-12b-cccg.pdf},
abstract = {Let $G$ be a cycle-free connected straight-line graph with
predefined edge lengths and fixed order of incident edges
around each vertex. We address the problem of deciding
whether there exists a simple polygon $P$ such that $G$ is
the straight skeleton of $P$. We show that for given $G$
such a polygon $P$ might not exist, and if it exists it
might not be unique. For the later case we give an example
with exponentially many suitable polygons. For small star
graphs and caterpillars we show necessary and sufficient
conditions for constructing $P$.\\ Considering only the
topology of the tree, that is, ignoring the length of the
edges, we show that any tree whose inner vertices have
degree at least $3$ is isomorphic to the straight skeleton
of a suitable convex polygon.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{acdhhlr-wmtss-12,
author = {O. Aichholzer and H.~Cheng and S.L.~Devadoss and T.~Hackl
and S.~Huber and B.~Li and A.~Risteski},
title = {{{What makes a Tree a Straight Skeleton?}}},
booktitle = {Proc. $28^{th}$ European Workshop on Computational
Geometry EuroCG '12},
pages = {137--140},
year = 2012,
address = {Assisi, Italy},
category = {3b},
thackl_label = {30C},
pdf = {/files/publications/geometry/acdhhlr-wmtss-12.pdf},
abstract = {Let $G$ be a cycle-free connected straight-line graph with
predefined edge lengths and fixed order of incident edges
around each vertex. We address the problem of deciding
whether there exists a simple polygon $P$ such that $G$ is
the straight skeleton of $P$. We show that for given $G$
such a polygon $P$ might not exist, and if it exists it
might not be unique. For small star graphs and caterpillars
we give necessary and sufficient conditions for
constructing $P$.},
originalfile = {/geometry/cggg.bib}
}
@article{afghhhuvv-okkps-15,
author = {Oswin Aichholzer and Ruy Fabila-Monroy and Hernan
Gonzalez-Aguilar and Thomas Hackl and Marco A. Heredia and
Clemens Huemer and Jorge Urrutia and Pavel Valtr and Birgit
Vogtenhuber},
title = {{{On $k$-Gons and $k$-Holes in Point Sets}}},
journal = {Computational Geometry: Theory and Applications},
year = 2015,
volume = {48},
number = {7},
pages = {528--537},
category = {3a},
thackl_label = {29J},
arxiv = {1409.0081},
doi = {http://dx.doi.org/10.1016/j.comgeo.2014.12.007},
pdf = {/files/publications/geometry/afghhhuvv-okkps-15.pdf},
abstract = {We consider a variation of the classical
Erd{\H{o}}os-Szekeres problems on the existence and number of
convex $k$-gons and $k$-holes (empty $k$-gons) in a set of
$n$ points in the plane. Allowing the $k$-gons to be
non-convex, we show bounds and structural results on
maximizing and minimizing their numbers. Most noteworthy,
for any $k$ and sufficiently large $n$, we give a quadratic
lower bound for the number of $k$-holes, and show that this
number is maximized by sets in convex position. We also
provide an improved lower bound for the number of convex
6-holes.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{afghhhuvv-okkps-11,
author = {Oswin Aichholzer and Ruy Fabila-Monroy and Hernan
Gonzalez-Aguilar and Thomas Hackl and Marco A. Heredia and
Clemens Huemer and Jorge Urrutia and Pavel Valtr and Birgit
Vogtenhuber},
title = {{{On $k$-Gons and $k$-Holes in Point Sets}}},
booktitle = {Proc. $23^{rd}$ Annual Canadian Conference on
Computational Geometry CCCG 2011},
pages = {21--26},
year = 2011,
address = {Toronto, Canada},
category = {3b},
thackl_label = {29C},
pdf = {/files/publications/geometry/afghhhuvv-okkps-11.pdf},
abstract = {We consider a variation of the classical
Erd{\H{o}}os-Szekeres problems on the existence and number of
convex $k$-gons and $k$-holes (empty $k$-gons) in a set of
$n$ points in the plane. Allowing the $k$-gons to be
non-convex, we show bounds and structural results on
maximizing and minimizing their numbers. Most noteworthy,
for any $k$ and sufficiently large $n$, we give a quadratic
lower bound for the number of $k$-holes, and show that this
number is maximized by sets in convex position. We also
provide an improved lower bound for the number of convex
6-holes.},
originalfile = {/geometry/cggg.bib}
}
@incollection{ahv-5g5h-12,
author = {O. Aichholzer and T. Hackl and B. Vogtenhuber},
title = {{{On 5-{G}ons and 5-{H}oles}}},
volume = {7579},
issue = {},
editor = {A.~Marquez and P.~Ramos and J.~Urrutia},
booktitle = {Computational Geometry: XIV Spanish Meeting on
Computational Geometry, EGC 2011, Festschrift
Dedicated to Ferran Hurtado on the Occasion of His
60th Birthday, Alcal{\'a} de Henares, Spain, June 27-30,
2011, Revised Selected Papers},
xxbooktitle = {Special issue: XIV Encuentros de Geometr\'{\i}a
Computacional ECG2011},
series = {Lecture Notes in Computer Science (LNCS)},
category = {3a},
thackl_label = {28J},
pdf = {/files/publications/geometry/ahv-5g5h-12.pdf},
pages = {1--13},
year = 2012,
publisher = {Springer},
abstract = {We consider an extension of a question of Erd{\H{o}}s on the
number of $k$-gons in a set of $n$ points in the plane.
Relaxing the convexity restriction we obtain results on
5-gons and 5-holes (empty 5-gons). In particular, we show a
direct relation between the number of non-convex 5-gons and
the rectilinear crossing number, provide an improved lower
bound for the number of convex 5-holes any point set must
contain, and prove that the number of general 5-holes is
asymptotically maximized for point sets in convex
position.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ahv-o5g5h-11,
author = {O. Aichholzer and T. Hackl and B. Vogtenhuber},
title = {{{On 5-gons and 5-holes}}},
booktitle = {Proc. XIV Encuentros de Geometr\'{\i}a Computacional},
category = {3b},
thackl_label = {28C},
pages = {7--10},
pdf = {/files/publications/geometry/ahv-o5g5h-11.pdf},
year = 2011,
address = {Alcal\'a, Spain},
abstract = {We consider an extention of a question of Erd{\H{o}}s on the
number of $k$-gons in a set of $n$ points in the plane.
Relaxing the convexity restriction we obtain results on
5-gons and 5-holes (empty 5-gons).},
originalfile = {/geometry/cggg.bib}
}
@article{afghhhuv-4hps-14,
author = {O.~Aichholzer and R.~Fabila-Monroy and
H.~Gonz{\'a}lez-Aguilar and T.~Hackl and M.A.~Heredia and
C.~Huemer and J.~Urrutia and B.~Vogtenhuber},
title = {{{\mbox{4-Holes} in Point Sets}}},
journal = {Computational Geometry: Theory and Applications},
note = {Special Issue on the 27th European Workshop on Computational Geometry (EuroCG 2011)},
year = 2014,
volume = {47},
number = {6},
pages = {644--650},
thackl_label = {27J},
category = {3a},
pdf = {/files/publications/geometry/afghhhuv-4hps-14.pdf},
doi = {http://dx.doi.org/10.1016/j.comgeo.2013.12.004},
abstract = {We consider a variant of a question of Erd{\H{o}}s on the
number of empty $k$-gons ($k$-holes) in a set of $n$ points
in the plane, where we allow the $k$-gons to be non-convex.
We show bounds and structural results on maximizing and
minimizing the number of general 4-holes, and maximizing
the number of non-convex 4-holes. In particular, we show
that for $n\geq 9$, the maximum number of general 4-holes
is ${n \choose 4}$, the minimum number of general 4-holes
is at least $\frac{5}{2}n^2 - \Theta(n)$, and the maximum
number of non-convex 4-holes is at least
$\frac{1}{2}n^3-\Theta(n^2\log n)$ and at most
$\frac{1}{2}n^3-\Theta(n^2)$.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{afghhhuv-4hps-11,
author = {O.~Aichholzer and R.~Fabila-Monroy and
H.~Gonz{\'a}lez-Aguilar and T.~Hackl and M.A.~Heredia and
C.~Huemer and J.~Urrutia and B.~Vogtenhuber},
title = {{{\mbox{4-Holes} in Point Sets}}},
booktitle = {Proc. $27^{th}$ European Workshop on Computational
Geometry EuroCG '11},
pages = {115--118},
year = 2011,
address = {Morschach, Switzerland},
thackl_label = {27C},
category = {3b},
pdf = {/files/publications/geometry/afghhhuv-4hps-11.pdf},
abstract = {We consider a variant of a question of Erd{\H{o}}s on the
number of empty $k$-gons ($k$-holes) in a set of $n$ points
in the plane, where we allow the $k$-gons to be non-convex.
We show bounds and structural results on maximizing and
minimizing the number of general \mbox{4-holes}, and
maximizing the number of non-convex \mbox{4-holes}.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aahw-emacc-11,
author = {O. Aichholzer and W. Aigner and T. Hackl and N. Wolpert},
title = {{{Exact medial axis computation for circular arc
boundaries}}},
booktitle = {Proc. $7^{th}$ International Conference on Curves and
Surfaces 2010 (Avignon, France), LNCS 6920},
editor = {J.D. Boissonat and M.L. Mazure and L.L. Schumaker},
address = {Avignon, France},
publisher = {Springer},
series = {Lecture Notes in Computer Science (LNCS)},
number = {6920},
category = {3b},
thackl_label = {26C},
pages = {28--42},
year = {2011},
pdf = {/files/publications/geometry/aahw-emacc-11.pdf},
abstract = {We propose a method to compute the algebraically correct
medial axis for simply connected planar domains which are
given by boundary representations composed of rational
circular arcs. The algorithmic approach is based on the
Divide-and-Conquer paradigm. However, we show how to avoid
inaccuracies in the medial axis computations arising from a
non-algebraic biarc construction of the boundary. To this
end we introduce the Exact Circular Arc Boundary
representation (ECAB), which allows algebraically exact
calculation of bisector curves. Fractions of these bisector
curves are then used to construct the exact medial axis. We
finally show that all necessary computations can be
performed over the fild of rational numbers with a small
number of adjoint square-roots.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{st10d,
author = {Oswin Aichholzer and Daniel Detassis and Thomas Hackl and
Gerald Steinbauer and Johannes Thonhauser},
title = {{"{Playing Pylos with an Autonomous Robot}"}},
booktitle = {IEEE/RSJ International Conference on Intelligent Robots
and Systems (IROS)},
address = {Taipei, Taiwan},
pages = {2507--2508},
category = {3b},
thackl_label = {25C},
pdf = {/files/publications/geometry/adhst-ppwar-10.pdf},
year = {2010},
abstract = {In this paper we present an autonomous robot which is able
to play the board game Pylos (Copyright by GIGAMIC s.a.
France) with a human opponent.},
alpha_part = {25C},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ahv-cppt-10,
author = {O. Aichholzer and T.~Hackl and B.~Vogtenhuber},
title = {{{Compatible Pointed Pseudo-Triangulations}}},
booktitle = {Proc. $22^{nd}$ Annual Canadian Conference on
Computational Geometry CCCG 2010},
pages = {91--94},
year = 2010,
address = {Winnipeg, Manitoba, Canada},
category = {3b},
thackl_label = {24C},
pdf = {/files/publications/geometry/ahv-cppt-10.pdf},
abstract = {For a given point set $S$ (in general position), two
pointed pseudo-triangulations are compatible if their union
is plane. We show that for any set $S$ there exist two
maximally disjoint compatible pointed
pseudo-triangulations, that is, their union is a
triangulation of~$S$. In contrast, we show that there are
point sets~$S$ and pointed pseudo-triangulations~$T$ such
that there exists no pointed pseudo-triangulation that is
compatible to and different from~$T$.},
originalfile = {/geometry/cggg.bib}
}
@article{afhkprv-bdt-12,
author = {O. Aichholzer and R.~Fabila-Monroy and T.~Hackl and M.~van
Kreveld and A.~Pilz and P.~Ramos and B.~Vogtenhuber},
title = {{{Blocking Delaunay Triangulations}}},
year = 2013,
volume = {46},
number = {2},
journal = {Computational Geometry: Theory and Applications},
pages = {154--159},
category = {3a},
oaich_label = {},
thackl_label = {23J},
doi = {http://dx.doi.org/10.1016/j.comgeo.2012.02.005},
pmcid = {3587385},
pmcurl = {http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3587385/},
pdf = {/files/publications/geometry/afhkprv-bdt-13.pdf},
abstract = {Given a set $B$ of $n$ blue points in general position, we
say that a set of red points $R$ blocks $B$ if in the
Delaunay triangulation of $B\cup R$ there is no edge
connecting two blue points. We give the following bounds
for the size of the smallest set $R$ blocking $B$:
(i)~$3n/2$ red points are always sufficient to block a set
of $n$ blue points, (ii)~if $B$ is in convex position,
$5n/4$ red points are always sufficient to block it, and
(iii)~at least $n-1$ red points are always necessary, and
there exist sets of blue points that require at least $n$
red points to be blocked.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{afhkprv-bdt-10,
author = {O. Aichholzer and R.~Fabila-Monroy and T.~Hackl and M.~van
Kreveld and A.~Pilz and P.~Ramos and B.~Vogtenhuber},
title = {{{Blocking Delaunay Triangulations}}},
booktitle = {Proc. $22^{nd}$ Annual Canadian Conference on
Computational Geometry CCCG 2010},
pages = {21--24},
year = 2010,
address = {Winnipeg, Manitoba, Canada},
category = {3b},
thackl_label = {23C},
pdf = {/files/publications/geometry/afhkprv-bdt-10.pdf},
abstract = {Given a set $B$ of $n$ blue points in general position, we
say that a set of red points $R$ blocks $B$ if in the
Delaunay triangulation of $B\cup R$ there is no edge
connecting two blue points. We give the following bounds
for the size of the smallest set $R$ blocking $B$:
(i)~$3n/2$ red points are always sufficient to block a set
of $n$ blue points, (ii)~if $B$ is in convex position,
$5n/4$ red points are always sufficient to block it, and
(iii)~at least $n-1$ red points are always necessary, and
there exist sets of blue points that require at least $n$
red points to be blocked.},
originalfile = {/geometry/cggg.bib}
}
@article{aahhpv-3cpt-15,
author = {O.~Aichholzer and F.~Aurenhammer and T. Hackl and
C.~Huemer and A.~Pilz and B.~Vogtenhuber},
title = {{{\mbox{3-Colorability} of Pseudo-Triangulations}}},
journal = {Int'l Journal of Computational Geometry \& Applications},
year = 2015,
volume = {25},
category = {3a},
number = {4},
oaich_label = {},
thackl_label = {22J},
pages = {283-298},
pdf = {/files/publications/geometry/aahhpv-3cpt-10.pdf},
doi = {http://dx.doi.org/10.1142/S0218195915500168},
xarxiv = {},
abstract = {Deciding $3$-colorability for general plane graphs is
known to be an NP-complete problem. However, for certain
classes of plane graphs, like triangulations, polynomial
time algorithms exist. We consider the family of
pseudo-triangulations (a generalization of triangulations)
and prove NP-completeness for this class, even if the
maximum face-degree is bounded to four, or pointed
pseudo-triangulations with maximum face degree five are
considered. As a complementary result, we show that for
pointed pseudo-triangulations with maximum face-degree
four, a $3$-coloring always exists and can be found in
linear time.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aahhpv-3cpt-10,
author = {O.~Aichholzer and F.~Aurenhammer and T. Hackl and
C.~Huemer and A.~Pilz and B.~Vogtenhuber},
title = {{{\mbox{3-Colorability} of Pseudo-Triangulations}}},
booktitle = {Proc. $26^{th}$ European Workshop on Computational
Geometry EuroCG '10},
pages = {21--24},
year = 2010,
address = {Dortmund, Germany},
thackl_label = {22C},
pdf = {/files/publications/geometry/aahhpv-3cpt-10.pdf},
abstract = {Deciding $3$-colorability for general plane graphs is
known to be an NP-complete problem. However, for certain
classes of plane graphs, like triangulations, polynomial
time algorithms exist. We consider the family of
pseudo-triangulations (a generalization of triangulations)
and prove NP-completeness for this class, even if the
maximum face-degree is bounded to four, or pointed
pseudo-triangulations with maximum face degree five are
considered. As a complementary result, we show that for
pointed pseudo-triangulations with maximum face-degree
four, a $3$-coloring always exists and can be found in
linear time.},
originalfile = {/geometry/cggg.bib}
}
@article{ahhprsv-pgpc-10,
author = {O. Aichholzer and T. Hackl and M. Hoffmann and A.~Pilz and
G.~Rote and B.~Speckmann and B.~Vogtenhuber},
title = {{{Plane graphs with parity constraints}}},
year = 2014,
volume = {30},
number = {1},
category = {3a},
journal = {Graphs and Combinatorics},
pdf = {/files/publications/geometry/ahhprsv-pgpc-12.pdf},
thackl_label = {20J},
pages = {47--69},
publisher = {Springer},
htmlnote = {For
Springer Online First.},
doi = {http://dx.doi.org/10.1007/s00373-012-1247-y},
abstract = {Let $S$ be a set of $n$ points in general position in the
plane. Together with $S$ we are given a set of parity
constraints, that is, every point of $S$ is labeled either
even or odd. A graph $G$ on $S$ satisfies the parity
constraint of a point $p \in S$, if the parity of the
degree of $p$ in $G$ matches its label. In this paper we
study how well various classes of planar graphs can satisfy
arbitrary parity constraints. Specifically, we show that we
can always find a plane tree, a two-connected outerplanar
graph, or a pointed pseudo-triangulation which satisfy all
but at most three parity constraints. With triangulations
we can satisfy about 2/3 of all parity constraints. In
contrast, for a given simple polygon $H$ with polygonal
holes on $S$, we show that it is NP-complete to decide
whether there exists a triangulation of $H$ that satisfies
all parity constraints.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ahhprsv-pgpc-09,
author = {O. Aichholzer and T. Hackl and M.~Hoffmann and A.~Pilz and
G.~Rote and B.~Speckmann and B.~Vogtenhuber},
title = {{{Plane Graphs with Parity Constraints}}},
pdf = {/files/publications/geometry/ahhprsv-pgpc-09.pdf},
oaich_label = {83},
thackl_label = {20C},
booktitle = {Lecture Notes in Computer Science (LNCS), Proc. $11^{th}$
International Workshop on Algorithms and Data Structures
(WADS)},
volume = {5664},
address = {Banff, Alberta, Canada},
pages = {13--24},
year = 2009,
abstract = {Let $S$ be a set of $n$ points in general position in the
plane. Together with $S$ we are given a set of parity
constraints, that is, every point of $S$ is labeled either
even or odd. A graph $G$ on $S$ satisfies the parity
constraint of a point $p \in S$, if the parity of the
degree of $p$ in $G$ matches its label. In this paper we
study how well various classes of planar graphs can satisfy
arbitrary parity constraints. Specifically, we show that we
can always find a plane tree, a two-connected outerplanar
graph, or a pointed pseudo-triangulation which satisfy all
but at most three parity constraints. With triangulations
we can satisfy about 2/3 of all parity constraints. In
contrast, for a given simple polygon $H$ with polygonal
holes on $S$, we show that it is NP-complete to decide
whether there exists a triangulation of $H$ that satisfies
all parity constraints.},
originalfile = {/geometry/cggg.bib}
}
@article{ahorrss-fgbdt-12,
author = {O.~Aichholzer and T.~Hackl and D.~Orden and P.~Ramos and
G.~Rote and A.~Schulz and B.~Speckmann},
title = {{{Flip Graphs of Bounded-Degree Triangulations}}},
journal = {Graphs and Combinatorics},
volume = {29},
number = {6},
pages = {1577--1593},
category = {3a},
pdf = {/files/publications/geometry/ahorrss-fgbdt-20120426.pdf},
thackl_label = {19J},
year = 2013,
eprint = {0903.2184},
archiveprefix = {arXiv},
htmlnote = {For
Springer Online First.},
doi = {http://dx.doi.org/10.1007/s00373-012-1229-0},
abstract = {We study flip graphs of triangulations whose maximum
vertex degree is bounded by a constant $k$. In particular,
we consider triangulations of sets of $n$ points in convex
position in the plane and prove that their flip graph is
connected if and only if $k > 6$; the diameter of the flip
graph is $O(n^2)$. We also show that, for general point
sets, flip graphs of pointed pseudo-triangulations can be
disconnected for $k \leq 9$, and flip graphs of
triangulations can be disconnected for any~$k$.
Additionally, we consider a relaxed version of the original
problem. We allow the violation of the degree bound $k$ by
a small constant. Any two triangulations with maximum
degree at most $k$ of a convex point set are connected in
the flip graph by a path of length $O(n \log n)$, where
every intermediate triangulation has maximum degree at most
$k+4$.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ahorrss-fgbdt-09,
author = {O.~Aichholzer and T.~Hackl and D.~Orden and P.~Ramos and
G.~Rote and A.~Schulz and B.~Speckmann},
title = {{{Flip Graphs of Bounded-Degree Triangulations}}},
booktitle = {Electronic Notes in Discrete Mathematics: Proc. European
Conference on Combinatorics, Graph Theory and Applications
EuroComb 2009},
volume = {34},
category = {3b},
pages = {509--513},
pdf = {/files/publications/geometry/ahorrss-fgbdt-09.pdf},
oaich_label = {84},
thackl_label = {19C},
year = 2009,
eprint = {0903.2184},
address = {Bordeaux, France},
abstract = {We study flip graphs of triangulations whose maximum
vertex degree is bounded by a constant $k$. Specifically,
we consider triangulations of sets of $n$ points in convex
position in the plane and prove that their flip graph is
connected if and only if $k > 6$; the diameter of the flip
graph is $O(n^2)$. We also show that for general point
sets, flip graphs of triangulations with degree $\leq k$
can be disconnected for any $k$.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aadhtv-lubne-09,
author = {O. Aichholzer and F. Aurenhammer and O.~Devillers and
T.~Hackl and M.~Teillaud and B.~Vogtenhuber},
title = {{{Lower and upper bounds on the number of empty
cylinders and ellipsoids}}},
booktitle = {Proc. $25^{th}$ European Workshop on Computational
Geometry EuroCG '09},
category = {3b},
pages = {139--142},
pdf = {/files/publications/geometry/aadhtv-lubne-09.pdf},
oaich_label = {79},
thackl_label = {18C},
year = 2009,
address = {Brussels, Belgium},
htmlnote = {Also available as Research Report RR-6748 "Counting
Quadrics and Delaunay Triangulations and a new Convex Hull
Theorem", INRIA, 2008, at
http://hal.inria.fr/inria-00343651.},
abstract = {Given a set $\cal S$ of $n$ points in three dimensions, we
study the maximum numbers of quadrics spanned by subsets of
points in $\cal S$ in various ways. Among various results
we prove that the number of empty circular cylinders is
between $\Omega(n^3)$ and $O(n^4)$ while we have a tight
bound $\Theta(n^4)$ for empty ellipsoids. We also take
interest in pairs of empty homothetic ellipsoids, with
application to the number of combinatorially distinct
Delaunay triangulations obtained by orthogonal projections
of $\cal S$ on a two-dimensional plane, which is
$\Omega(n^4)$ and $O(n^5)$.
A side result is that the convex hull in $d$ dimensions of
a set of $n$ points, where one half lies in a subspace of
odd dimension~\mbox{$\delta > \frac{d}{2}$}, and the second
half is the (multi-dimensional) projection of the first
half on another subspace of dimension~$\delta$, has
complexity only $O\left(n^{\frac{d}{2}-1}\right)$.},
originalfile = {/geometry/cggg.bib}
}
@article{ahhhv-lbpsa-09b,
author = {O.~Aichholzer and T.~Hackl and C.~Huemer and F.~Hurtado
and B.~Vogtenhuber},
title = {{{Large bichromatic point sets admit empty monochromatic $4$-gons}}},
year = 2010,
journal = {SIAM Journal on Discrete Mathematics (SIDMA)},
volume = {23},
number = {4},
pages = {2147--2155},
category = {3a},
doi = {http://dx.doi.org/10.1137/090767947},
oaich_label = {78b},
thackl_label = {17J},
pdf = {/files/publications/geometry/ahhhv-lbpsa-09b.pdf},
abstract = {We consider a variation of a problem stated by {Erd\"os}
and Szekeres in 1935 about the existence of a number
$f^\textrm{ES}(k)$ such that any set $S$ of at least
$f^\textrm{ES}(k)$ points in general position in the plane
has a subset of $k$ points that are the vertices of a
convex $k$-gon. In our setting the points of $S$ are
colored, and we say that a (not necessarily convex) spanned
polygon is monochromatic if all its vertices have the same
color. Moreover, a polygon is called empty if it does not
contain any points of $S$ in its interior. We show that any
bichromatic set of $n \geq 5044$ points in $\mathcal{R}^2$
in general position determines at least one empty,
monochromatic quadrilateral (and thus linearly many).},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ahhhv-lbpsa-09,
author = {O.~Aichholzer and T.~Hackl and C.~Huemer and F.~Hurtado
and B.~Vogtenhuber},
title = {{{Large bichromatic point sets admit empty monochromatic $4$-gons}}},
booktitle = {Proc. $25^{th}$ European Workshop on Computational
Geometry EuroCG '09},
category = {3b},
pages = {133--136},
pdf = {/files/publications/geometry/ahhhv-lbpsa-09.pdf},
oaich_label = {78},
thackl_label = {17C},
year = 2009,
address = {Brussels, Belgium},
abstract = {We consider a variation of a problem stated by Erd\"os and
Szekeres in 1935 about the existence of a number
$f^\textrm{ES}(k)$ such that any set $S$ of at least
$f^\textrm{ES}(k)$ points in general position in the plane
has a subset of $k$ points that are the vertices of a
convex $k$-gon. In our setting the points of $S$ are
colored, and we say that a (not necessarily convex) spanned
polygon is monochromatic if all its vertices have the same
color. Moreover, a polygon is called empty if it does not
contain any points of $S$ in its interior. We show that any
bichromatic set of $n \geq 5044$ points in $\mathcal{R}^2$
in general position determines at least one empty,
monochromatic quadrilateral (and thus linearly many).},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{affhhuv-mimp-09,
author = {O. Aichholzer and R.~Fabila-Monroy and
D.~Flores-Pe{\~n}aloza and T.~Hackl and C.~Huemer and
J.~Urrutia and B.~Vogtenhuber},
title = {{{Modem Illumination of Monotone Polygons}}},
booktitle = {Proc. $25^{th}$ European Workshop on Computational
Geometry EuroCG '09},
category = {3b},
pages = {167--170},
arxiv = {1503.05062},
pdf = {/files/publications/geometry/affhhuv-mimp-09.pdf},
oaich_label = {80},
thackl_label = {16C},
year = 2009,
address = {Brussels, Belgium},
abstract = {We study a generalization of the classical problem of
illumination of polygons. Instead of modeling a light
source we model a wireless device whose radio signal can
penetrate a given number $k$ of walls. We call these
objects $k$-modems and study the minimum number of
$k$-modems necessary to illuminate monotone and monotone
orthogonal polygons. We show that every monotone polygon on
$n$ vertices can be illuminated with $\left\lceil
\frac{n}{2k} \right\rceil$ $k$-modems and exhibit examples
of monotone polygons requiring $\left\lceil \frac{n}{2k+2}
\right\rceil$ $k$-modems. For monotone orthogonal polygons,
we show that every such polygon on $n$ vertices can be
illuminated with $\left\lceil \frac{n}{2k+4} \right\rceil$
$k$-modems and give examples which require $\left\lceil
\frac{n}{2k+4} \right\rceil$ $k$-modems for $k$ even and
$\left\lceil \frac{n}{2k+6} \right\rceil$ for $k$ odd.},
originalfile = {/geometry/cggg.bib}
}
@article{aaahjpr-dcvdr-10,
author = {O. Aichholzer and W. Aigner and F. Aurenhammer and
T.~Hackl and B.~J\"uttler and E.~Pilgerstorfer and M.~Rabl},
title = {{{Divide-and conquer for {V}oronoi diagrams revisited}}},
journal = {Computational Geometry: Theory and Applications},
note = {Special Issue on the 25th Annual Symposium on
Computational Geometry (SoCG'09)},
pages = {688--699},
volume = {43},
number = {8},
category = {3a},
doi = {http://dx.doi.org/10.1016/j.comgeo.2010.04.004},
year = 2010,
pdf = {/files/publications/geometry/aaahjpr-dcvdr-09b.pdf},
thackl_label = {15J},
abstract = {We show how to divide the edge graph of a Voronoi diagram
into a tree that corresponds to the medial axis of an
(augmented) planar domain. Division into base cases is then
possible, which, in the bottom-up phase, can be merged by
trivial concatenation. The resulting construction
algorithm---similar to Delaunay triangulation methods---is
not bisector-based and merely computes dual links between
the sites, its atomic steps being inclusion tests for sites
in circles. This guarantees computational simplicity and
numerical stability. Moreover, no part of the Voronoi
diagram, once constructed, has to be discarded again. The
algorithm works for polygonal and curved objects as sites
and, in particular, for circular arcs which allows its
extension to general free-form objects by Voronoi diagram
preserving and data saving biarc approximations. The
algorithm is randomized, with expected runtime $O(n\log n)$
under certain assumptions on the input data. Experiments
substantiate an efficient behavior even when these
assumptions are not met. Applications to offset
computations and motion planning for general objects are
described.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aaahjpr-dcvdr-09b,
author = {O. Aichholzer and W.~Aigner and F. Aurenhammer and
T.~Hackl and B.~J{\"u}ttler and E.~Pilgerstorfer and
M.~Rabl},
title = {{{Divide-and-Conquer for Voronoi Diagrams Revisited}}},
booktitle = {$25^{th}$ Ann. ACM Symp. Computational Geometry},
category = {3b},
pages = {189--197},
pdf = {/files/publications/geometry/aaahjpr-dcvdr-09b.pdf},
oaich_label = {81b},
thackl_label = {15C},
year = 2009,
address = {Aarhus, Denmark},
abstract = {We propose a simple and practical divide-and-conquer
algorithm for constructing planar Voronoi diagrams. The
novel aspect of the algorithm is its emphasis on the
top-down phase, which makes it applicable to sites of
general shape.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aaahjpr-dcvdr-09,
author = {O. Aichholzer and W.~Aigner and F. Aurenhammer and
T.~Hackl and B.~J{\"u}ttler and E.~Pilgerstorfer and
M.~Rabl},
title = {{{Divide-and-Conquer for Voronoi Diagrams Revisited}}},
booktitle = {Proc. $25^{th}$ European Workshop on Computational
Geometry EuroCG '09},
category = {3b},
pages = {293--296},
pdf = {/files/publications/geometry/aaahjpr-dcvdr-09.pdf},
oaich_label = {81},
thackl_label = {15C},
year = 2009,
address = {Brussels, Belgium},
abstract = {We propose a simple and practical divide-and-conquer
algorithm for constructing planar Voronoi diagrams. The
novel aspect of the algorithm is its emphasis on the
top-down phase, which makes it applicable to sites of
general shape.},
originalfile = {/geometry/cggg.bib}
}
@article{affhhj-emt-09,
author = {O. Aichholzer and R. Fabila-Monroy and D.
Flores-Pe{\~n}aloza and T.~Hackl and C.~Huemer and
J.~Urrutia},
title = {{{Empty Monochromatic Triangles}}},
year = 2009,
journal = {Computational Geometry: Theory and Applications},
volume = {42},
number = {9},
pages = {934--938},
doi = {http://dx.doi.org/10.1016/j.comgeo.2009.04.002},
category = {3a},
oaich_label = {75b},
thackl_label = {14J},
pdf = {/files/publications/geometry/affhhj-emt-09.pdf},
abstract = {We consider a variation of a problem stated by Erd\"os and
Guy in 1973 about the number of convex $k$-gons determined
by any set $S$ of $n$ points in the plane. In our setting
the points of $S$ are colored and we say that a spanned
polygon is monochromatic if all its points are colored with
the same color. As a main result we show that any
bi-colored set of $n$ points in $\mathcal{R}^2$ in general
position determines a super-linear number of empty
monochromatic triangles, namely $\Omega(n^{5/4})$.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{affhhj-emt-08,
author = {O. Aichholzer and R. Fabila-Monroy and D.
Flores-Pe{\~n}aloza and T.~Hackl and C.~Huemer and
J.~Urrutia},
title = {{{Empty Monochromatic Triangles}}},
booktitle = {Proc. $20^{th}$ Annual Canadian Conference on
Computational Geometry CCCG 2008},
pages = {75--78},
year = 2008,
address = {Montreal, Quebec, Canada},
category = {3b},
oaich_label = {75},
thackl_label = {14C},
pdf = {/files/publications/geometry/affhhj-emt-08.pdf},
abstract = {We consider a variation of a problem stated by Erd\"os and
Guy in 1973 about the number of convex $k$-gons determined
by any set $S$ of $n$ points in the plane. In our setting
the points of $S$ are colored and we say that a spanned
polygon is monochromatic if all its points are colored with
the same color. As a main result we show that any
bi-colored set of $n$ points in $\mathcal{R}^2$ in general
position determines a super-linear number of empty
monochromatic triangles, namely $\Omega(n^{5/4})$.},
originalfile = {/geometry/cggg.bib}
}
@article{acffhhhw-erncc-09,
author = {O. Aichholzer and S. Cabello and R. Fabila-Monroy and D.
Flores-Pe{\~n}aloza and T.~Hackl and C.~Huemer and
F.~Hurtado and D.R.~Wood},
title = {{{Edge-Removal and Non-Crossing Configurations in Geometric Graphs}}},
journal = {Discrete Mathematics \& Theoretical Computer Science
(DMTCS)},
year = 2010,
volume = {12},
category = {3a},
number = {1},
oaich_label = {83},
thackl_label = {13J},
pages = {75--86},
pdf = {/files/publications/geometry/acffhhhw-erncc-09.pdf},
abstract = {We study the {following} extremal problem for geometric
graphs: How many arbitrary edges can be removed from a
complete geometric graph with $n$ vertices such that the
remaining graph still contains a certain non-crossing
subgraph. In particular we consider perfect matchings and
subtrees of a given size. For both classes of geometric
graphs we obtain tight bounds on the maximum number of
removable edges. We further present several conjectures and
bounds on the number of removable edges for other classes
of non-crossing geometric graphs.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{acffhhhw-erncc-08,
author = {O. Aichholzer and S. Cabello and R. Fabila-Monroy and D.
Flores-Pe{\~n}aloza and T.~Hackl and C.~Huemer and
F.~Hurtado and D.R.~Wood},
title = {{{Edge-Removal and Non-Crossing Configurations in Geometric Graphs}}},
booktitle = {Proc. $24^{th}$ European Workshop on Computational
Geometry EuroCG '08},
pages = {119--122},
pdf = {/files/publications/geometry/acffhhhw-erncc-08.pdf},
oaich_label = {73},
thackl_label = {13C},
year = 2008,
address = {Nancy, France},
abstract = {We study the following extremal problem for geometric
graphs: How many arbitrary edges can be removed from a
complete geometric graph with $n$ vertices such that the
remaining graph still contains a certain non-crossing
subgraph. In particular we consider perfect matchings and
subtrees of a given size. For both classes of geometric
graphs we obtain tight bounds on the maximum number of
removable edges. We further present several conjectures and
bounds on the number of removable edges for other classes
of non-crossing geometric graphs.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aahkprsv-spia-08,
author = {O. Aichholzer and F. Aurenhammer and T.~Hackl and
B.~Kornberger and S.~Plantinga and G.~Rote and A.~Sturm and
G.~Vegter},
title = {{{Seed Polytopes for Incremental Approximation}}},
booktitle = {Proc. $24^{th}$ European Workshop on Computational
Geometry EuroCG '08},
pages = {13--16},
postscript = {/files/publications/geometry/aahkprsv-spia-08.ps.gz},
oaich_label = {74},
thackl_label = {12C},
year = 2008,
address = {Nancy, France},
abstract = {Approximating a given three-dimensional object in order to
simplify its handling is a classical topic in computational
geometry and related fields. A typical approach is based on
incremental approximation algorithms, which start with a
small and topologically correct polytope representation
(the seed polytope) of a given sample point cloud or input
mesh. In addition, a correspondence between the faces of
the polytope and the respective regions of the object
boundary is needed to guarantee correctness.
We construct such a polytope by first computing a
simplified though still homotopy equivalent medial axis
transform of the input object. Then, we inflate this medial
axis to a polytope of small size. Since our approximation
maintains topology, the simplified medial axis transform is
also useful for skin surfaces and envelope surfaces.},
originalfile = {/geometry/cggg.bib}
}
@article{aaahjr-macpf-08,
author = {O. Aichholzer and W. Aigner and F. Aurenhammer and
T.~Hackl and B.~J{\"u}ttler and M.~Rabl},
title = {{{Medial Axis Computation for Planar Free-Form Shapes}}},
journal = {Computer-Aided Design},
note = {Special issue: {V}oronoi Diagrams and their Applications},
year = 2009,
volume = {41},
category = {3a},
number = {5},
oaich_label = {77},
thackl_label = {11J},
doi = {http://dx.doi.org/10.1016/j.cad.2008.08.008},
pdf = {/files/publications/geometry/aaahjr-macpf-09.pdf},
pages = {339--349},
abstract = {We present a simple, efficient, and stable method for
computing---with any desired precision---the medial axis of
simply connected planar domains. The domain boundaries are
assumed to be given as polynomial spline curves. Our
approach combines known results from the field of geometric
approximation theory with a new algorithm from the field of
computational geometry. Challenging steps are (1) the
approximation of the boundary spline such that the medial
axis is geometrically stable, and (2) the efficient
decomposition of the domain into base cases where the
medial axis can be computed directly and exactly. We solve
these problems via spiral biarc approximation and a
randomized divide \& conquer algorithm.},
originalfile = {/geometry/cggg.bib}
}
@proceedings{ah-pewcg-07,
author = {O. Aichholzer and T. Hackl},
editor = {O. Aichholzer and T. Hackl},
title = {{{Collection of Abstracts of the $23^{rd}$ European Workshop
on Computational Geometry 2007}}},
booktitle = {{{Collection of Abstracts of the $23^{rd}$ European Workshop
on Computational Geometry 2007}}},
pages = {1--254},
oaich_label = {69},
thackl_label = {10P},
year = {2007},
address = {Graz, Austria},
isbn = {978-3-902465-62-7},
htmlnote = {Available at the conference homepage http://ewcg07.tugraz.at/EuroCG2007Abstracts.pdf.},
abstract = {The {\bf $\mathbf{ 23^{rd}}$ European Workshop on
Computational Geometry} (EWCG'07) was held at the
University of Technology in Graz (Austria) on March
$19^{th} - 21^{st}$, 2007. More information about the
workshop can be found at {\tt http://ewcg07.tugraz.at}.
This collection of extended abstracts contains the $60$
scientific contributions as well as three invited talks
presented at the workshop. The submission record of over
$70$ abstracts from more than $20$ different countries,
covering a wide range of topics, shows that Computational
Geometry is a lively and still growing research field in
Europe.},
originalfile = {/geometry/cggg.bib}
}
@article{ahhhssv-mmaps-09,
author = {O. Aichholzer and T. Hackl and M.~Hoffmann and C.~Huemer
and A.~P{\'o}r and F.~Santos and B.~Speckmann and B.~Vogtenhuber},
title = {{{Maximizing Maximal Angles for Plane Straight Line Graphs}}},
year = 2013,
volume = {46},
number = {1},
journal = {Computational Geometry: Theory and Applications},
pages = {17--28},
category = {3a},
oaich_label = {65c},
thackl_label = {9J},
pdf = {/files/publications/geometry/ahhhpssv-mmaps-12.pdf},
doi = {http://dx.doi.org/10.1016/j.comgeo.2012.03.002},
eprint = {0705.3820},
archiveprefix = {arXiv},
abstract = {Let $G=(S, E)$ be a plane straight line graph on a finite
point set $S\subset {R}^2$ in general position. For a point
$p\in S$ let the {\em maximum incident angle} of $p$ in $G$
be the maximum angle between any two edges of $G$ that
appear consecutively in the circular order of the edges
incident to $p$. A plane straight line graph is called {\em
$\varphi$-open} if each vertex has an incident angle of
size at least $\varphi$. In this paper we study the
following type of question: What is the maximum angle
$\varphi$ such that for any finite set $S\subset {R}^2$ of
points in general position we can find a graph from a
certain class of graphs on $S$ that is $\varphi$-open? In
particular, we consider the classes of triangulations,
spanning trees, and paths on $S$ and give tight bounds in
all but one cases.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ahhhssv-mmaps-07b,
author = {O. Aichholzer and T. Hackl and M.~Hoffmann and C.~Huemer
and A.~Por and F.~Santos and B.~Speckmann and
B.~Vogtenhuber},
title = {{{Maximizing Maximal Angles for Plane Straight Line Graphs}}},
pdf = {/files/publications/geometry/ahhhssv-mmaps-07b.pdf},
oaich_label = {65b},
thackl_label = {9C},
booktitle = {Lecture Notes in Computer Science (LNCS), Proc. $10^{th}$
International Workshop on Algorithms and Data Structures
(WADS)},
volume = {4619},
address = {Halifax, Nova Scotia, Canada},
pages = {458--469},
year = 2007,
eprint = {0705.3820},
doi = {10.1007/978-3-540-73951-7_40},
abstract = {Let $G=(S, E)$ be a plane straight line graph on a finite
point set $S\subset {R}^2$ in general position. For a point
$p\in S$ let the {\em maximum incident angle} of $p$ in $G$
be the maximum angle between any two edges of $G$ that
appear consecutively in the circular order of the edges
incident to $p$. A plane straight line graph is called {\em
$\varphi$-open} if each vertex has an incident angle of
size at least $\varphi$. In this paper we study the
following type of question: What is the maximum angle
$\varphi$ such that for any finite set $S\subset {R}^2$ of
points in general position we can find a graph from a
certain class of graphs on $S$ that is $\varphi$-open? In
particular, we consider the classes of triangulations,
spanning trees, and paths on $S$ and give tight bounds in
all but one cases.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ahhhssv-mmaps-07,
author = {O. Aichholzer and T.~Hackl and M.~Hoffmann and C.~Huemer
and F.~Santos and B.~Speckmann and B.~Vogtenhuber},
title = {{{Maximizing Maximal Angles for Plane Straight Line Graphs}}},
booktitle = {Proc. $23^{rd}$ European Workshop on Computational
Geometry EuroCG '07},
pages = {98--101},
pdf = {/files/publications/geometry/ahhhssv-mmaps-07.pdf},
oaich_label = {65},
thackl_label = {9C},
year = 2007,
address = {Graz, Austria},
eprint = {0705.3820},
archiveprefix = {arXiv},
htmlnote = {Also available as FSP-report S092-48, Austria, 2007, at http://www.industrial-geometry.at/techrep.php.},
abstract = {Let $G=(S, E)$ be a plane straight line graph on a finite
point set $S\subset {R}^2$ in general position. For a point
$p\in S$ let the {\em maximum incident angle} of $p$ in $G$
be the maximum angle between any two edges of $G$ that
appear consecutively in the circular order of the edges
incident to $p$. A plane straight line graph is called {\em
$\varphi$-open} if each vertex has an incident angle of
size at least $\varphi$. In this paper we study the
following type of question: What is the maximum angle
$\varphi$ such that for any finite set $S\subset {R}^2$ of
points in general position we can find a graph from a
certain class of graphs on $S$ that is $\varphi$-open? In
particular, we consider the classes of triangulations,
spanning trees, and paths on $S$ and give tight bounds in
all but one cases.},
originalfile = {/geometry/cggg.bib}
}
@article{aahjos-csacb-09,
author = {O. Aichholzer and F. Aurenhammer and T.~Hackl and
B.~J\"uttler and M.~Oberneder and Z.~S\'ir},
title = {{{Computational and Structural Advantages of Circular Boundary Representation}}},
oaich_label = {68b},
thackl_label = {8J},
year = 2011,
volume = {21},
number = {1},
journal = {Int'l. Journal of Computational Geometry \& Applications},
pages = {47--69},
category = {3a},
pdf = {/files/publications/geometry/aahjos-csacb-09.pdf},
abstract = {Boundary approximation of planar shapes by circular arcs
has quantitive and qualitative advantages compared to using
straight-line segments. We demonstrate this by way of three
basic and frequent computations on shapes -- convex hull,
decomposition, and medial axis. In particular, we propose a
novel medial axis algorithm that beats existing methods in
simplicity and practicality, and at the same time
guarantees convergence to the medial axis of the original
shape.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aahjos-csacb-07,
author = {O. Aichholzer and F. Aurenhammer and T.~Hackl and
B.~J\"uttler and M.~Oberneder and Z.~S\'ir},
title = {{{Computational and Structural Advantages of Circular
Boundary Representation}}},
oaich_label = {68},
thackl_label = {8C},
booktitle = {Lecture Notes in Computer Science (LNCS), Proc. $10^{th}$
International Workshop on Algorithms and Data Structures
(WADS)},
volume = {4619},
address = {Halifax, Nova Scotia, Canada},
pages = {374--385},
year = 2007,
category = {3b},
postscript = {/files/publications/geometry/aahjos-csacb-07.ps.gz},
htmlnote = {Also available as FSP-report S092-38, Austria, 2006, at http://www.industrial-geometry.at/techrep.php.},
abstract = {Boundary approximation of planar shapes by circular arcs
has quantitive and qualitative advantages compared to using
straight-line segments. We demonstrate this by way of three
basic and frequent computations on shapes -- convex hull,
decomposition, and medial axis. In particular, we propose a
novel medial axis algorithm that beats existing methods in
simplicity and practicality, and at the same time
guarantees convergence to the medial axis of the original
shape.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aahkpp-abtob-07,
author = {O. Aichholzer and F. Aurenhammer and T.~Hackl and
B.~Kornberger and M.~Peternell and H.~Pottmann},
title = {{{Approximating Boundary-Triangulated Objects with Balls}}},
booktitle = {Proc. $23^{rd}$ European Workshop on Computational
Geometry EuroCG '07},
pages = {130--133},
pdf = {/files/publications/geometry/aahkpp-abtop-07.pdf},
oaich_label = {66},
thackl_label = {7C},
year = 2007,
address = {Graz, Austria},
htmlnote = {Also available as FSP-report S092-49, Austria, 2007, at http://www.industrial-geometry.at/techrep.php.},
abstract = {We compute a set of balls that approximates a given
\mbox{3D object}, and we derive small additive bounds for
the overhead in balls with respect to the minimal solution
with the same quality. The algorithm has been implemented
and tested using the CGAL library.},
originalfile = {/geometry/cggg.bib}
}
@article{aahs-mwpt-08,
author = {O. Aichholzer and F. Aurenhammer and T.~Hackl and
B.~Speckmann},
title = {{{On Minimum Weight Pseudo-Triangulations}}},
journal = {Computational Geometry: Theory and Applications},
pages = {627--631},
volume = {42},
number = {6-7},
category = {3a},
oaich_label = {71a},
thackl_label = {6J},
year = 2009,
pdf = {/files/publications/geometry/aahs-mwpt-09.pdf},
abstract = {In this note we discuss some structural properties of
minimum weight pseudo-triangulations of point sets.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aahs-pmwpt-07,
author = {O. Aichholzer and F. Aurenhammer and T.~Hackl and
B.~Speckmann},
title = {{{On (Pointed) Minimum Weight Pseudo-Triangulations}}},
booktitle = {Proc. $19^{th}$ Annual Canadian Conference on
Computational Geometry CCCG 2007},
pages = {209--212},
year = 2007,
address = {Ottawa, Ontario, Canada},
category = {3b},
oaich_label = {71},
thackl_label = {6C},
pdf = {/files/publications/geometry/aahs-pmwpt-07.pdf},
abstract = {In this note we discuss some structural properties of
minimum weight (pointed) pseudo-triangulations.},
originalfile = {/geometry/cggg.bib}
}
@article{aahh-ccps-06,
author = {O.~Aichholzer and F.~Aurenhammer and T.~Hackl and
C.~Huemer},
title = {{{Connecting Colored Point Sets}}},
journal = {Discrete Applied Mathematics},
year = 2007,
volume = {155},
number = {3},
pages = {271--278},
category = {3a},
oaich_label = {60},
thackl_label = {5J},
postscript = {/files/publications/geometry/aahh-ccps-06.ps.gz},
htmlnote = {Also available as FSP-report S092-45, Austria, 2006, at http://www.industrial-geometry.at/techrep.php.},
abstract = {We study the following Ramsey-type problem. Let \mbox{$S =
B \cup R$} be a two-colored set of $n$ points in the plane.
We show how to construct, in \mbox{$O(n \log n)$} time, a
crossing-free spanning tree $T(R)$ for~$R$, and a
crossing-free spanning tree $T(B)$ for~$B$, such that both
the number of crossings between $T(R)$ and $T(B)$ and the
diameters of~$T(R)$ and $T(B)$ are kept small. The
algorithm is conceptually simple and is implementable
without using any non-trivial data structure. This improves
over a previous method in Tokunaga~\cite{T} that is less
efficient in implementation and does not guarantee a
diameter bound. },
originalfile = {/geometry/cggg.bib}
}
@article{aah-ptlc-06a,
author = {O. Aichholzer and F. Aurenhammer and T.~Hackl},
title = {{{Pre-triangulations and liftable complexes}}},
journal = {Discrete \& Computational Geometry},
year = 2007,
volume = {38},
category = {3a},
number = {},
oaich_label = {61a},
thackl_label = {4J},
pages = {701--725},
postscript = {/files/publications/geometry/aah-ptlc-06a.ps.gz},
abstract = {We introduce the concept of pre-triangulations, a
relaxation of triangulations that goes beyond the
frequently used concept of pseudo-triangulations.
Pre-triangulations turn out to be more natural than
pseudo-triangulations in certain cases. We show that
pre-triangulations arise in three different contexts: In
the characterization of polygonal complexes that are
liftable to three-space in a strong sense, in flip
sequences for general polygonal complexes, and as graphs of
maximal locally convex functions.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aah-ptlc-06,
author = {O. Aichholzer and F. Aurenhammer and T.~Hackl},
title = {{{Pre-triangulations and liftable complexes}}},
booktitle = {$22^{nd}$ Ann. ACM Symp. Computational Geometry},
year = 2006,
pages = {282--291},
category = {3b},
oaich_label = {61},
thackl_label = {4C},
address = {Sedona, Arizona, USA},
postscript = {/files/publications/geometry/aah-ptlc-06.ps.gz},
htmlnote = {Also available as FSP-report S092-6, Austria, 2006, at http://www.industrial-geometry.at/techrep.php.},
abstract = {We introduce and discuss the concept of
pre-triangulations, a relaxation of triangulations that
goes beyond the well-established class of
pseudo-triangulations.},
originalfile = {/geometry/cggg.bib}
}
@article{ahhhkv-npg-06a,
author = {O. Aichholzer and T. Hackl and C.~Huemer and F.~Hurtado
and H.~Krasser and B.~Vogtenhuber},
title = {{{On the number of plane geometric graphs}}},
journal = {Graphs and Combinatorics (Springer)},
pages = {67--84},
volume = {23(1)},
oaich_label = {58a},
thackl_label = {3J},
category = {3a},
postscript = {/files/publications/geometry/ahhhkv-npgg-06.ps.gz},
year = 2007,
doi = {https://doi.org/10.1007/s00373-007-0704-5},
abstract = {We investigate the number of plane geometric, i.e.,
straight-line, graphs, a set $S$ of $n$ points in the plane
admits. We show that the number of plane geometric graphs
and connected plane geometric graphs as well as the number
of cycle-free plane geometric graphs is minimized when $S$
is in convex position. Moreover, these results hold for all
these graphs with an arbitrary but fixed number of edges.
Consequently, we provide a unified proof that the
cardinality of any family of acyclic graphs (for example
spanning trees, forests, perfect matchings, spanning paths,
and more) is minimized for point sets in convex position.
In addition we construct a new extremal configuration, the
so-called double zig-zag chain. Most noteworthy this
example bears $\Theta^*(\sqrt{72}\,^n)$ =
$\Theta^*(8.4853^n)$ triangulations and
$\Theta^*(41.1889^n)$ plane graphs (omitting polynomial
factors in both cases), improving the previously known best
maximizing examples.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ahhhkv-npg-06,
author = {O. Aichholzer and T. Hackl and C.~Huemer and F.~Hurtado
and H.~Krasser and B.~Vogtenhuber},
title = {{{On the number of plane graphs}}},
booktitle = {Proc. $17^{th}$ Annual ACM-SIAM Symposium on Discrete
Algorithms (SODA)},
pages = {504-513},
year = 2006,
address = {Miami, Florida, USA},
category = {3b},
oaich_label = {58},
thackl_label = {3C},
pdf = {/files/publications/geometry/ahhhkv-npg-06.pdf},
htmlnote = {Also available as FSP-report S092-8, Austria, 2006, at http://www.industrial-geometry.at/techrep.php.},
abstract = {We investigate the number of plane geometric, i.e.,
straight-line, graphs, a set $S$ of $n$ points in the plane
admits. We show that the number of plane graphs and
connected plane graphs as well as the number of cycle-free
plane graphs is minimized when $S$ is in convex position.
Moreover, these results hold for all these graphs with an
arbitrary but fixed number of edges. Consequently, we
provide simple proofs that the number of spanning trees,
cycle-free graphs (forests), perfect matchings, and
spanning paths is also minimized for point sets in convex
position. In addition we construct a new extremal
configuration, the so-called double zig-zag chain. Most
noteworthy this example bears $\Theta^*(\sqrt{72}\,^n)$ =
$\Theta^*(8.4853^n)$ triangulations and
$\Theta^*(41.1889^n)$ plane graphs (omitting polynomial
factors in both cases), improving the previously known best
maximizing examples.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ahhhkv-bnpg-06,
author = {O. Aichholzer and T. Hackl and C.~Huemer and F.~Hurtado
and H.~Krasser and B.~Vogtenhuber},
title = {{{Bounding the number of plane graphs}}},
booktitle = {Proc. $15^{th}$ Annual Fall Workshop on Computational
Geometry and Visualization},
pages = {31-32},
year = 2005,
address = {Philadelphia, Pennsylvania, USA},
category = {3b},
oaich_label = {58b},
thackl_label = {3C},
abstract = {We investigate the number of plane geometric, i.e.,
straight-line, graphs, a set $S$ of $n$ points in the plane
admits. We show that the number of plane graphs and
connected plane graphs as well as the number of cycle-free
plane graphs is minimized when $S$ is in convex position.
Moreover, these results hold for all these graphs with an
arbitrary but fixed number of edges. Consequently, we
provide simple proofs that the number of spanning trees,
cycle-free graphs (forests), perfect matchings, and
spanning paths is also minimized for point sets in convex
position. In addition we construct a new extremal
configuration, the so-called double zig-zag chain. Most
noteworthy this example bears $\Theta^*(\sqrt{72}\,^n)$ =
$\Theta^*(8.4853^n)$ triangulations and
$\Theta^*(41.1889^n)$ plane graphs (omitting polynomial
factors in both cases), improving the previously known best
maximizing examples.},
originalfile = {/geometry/cggg.bib}
}
@article{aaghhhkrv-mefpp-06,
author = {O. Aichholzer and F. Aurenhammer and P. Gonzalez-Nava and
T.~Hackl and C.~Huemer and F.~Hurtado and H.~Krasser and
S.~Ray and B.~Vogtenhuber},
title = {{{Matching Edges and Faces in Polygonal Partitions}}},
journal = {Computational Geometry: Theory and Applications},
pages = {134--141},
volume = {39(2)},
category = {3a},
oaich_label = {57a},
thackl_label = {2J},
year = 2008,
postscript = {/files/publications/geometry/aaghhhkrv-mefpp-06.ps.gz},
abstract = {We define general Laman (count) conditions for edges and
faces of polygonal partitions in the plane. Several
well-known classes, including $k$-regular partitions,
$k$-angulations, and rank $k$ pseudo-triangulations, are
shown to fulfill such conditions. As a consequence
non-trivial perfect matchings exist between the edge sets
(or face sets) of two such structures when they live on the
same point set. We also describe a link to spanning tree
decompositions that applies to quadrangulations and certain
pseudo-triangulations.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aaghhhkrv-mefpp-05,
author = {O. Aichholzer and F. Aurenhammer and P.~Gonzalez-Nava and
T.~Hackl and C.~Huemer and F.~Hurtado and H.~Krasser and
S.~Ray and B.~Vogtenhuber},
title = {{{Matching Edges and Faces in Polygonal Partitions}}},
booktitle = {Proc. $17^{th}$ Annual Canadian Conference on
Computational Geometry CCCG 2005},
pages = {123--126},
year = 2005,
address = {Windsor, Ontario, Canada},
category = {3b},
oaich_label = {57},
thackl_label = {2C},
postscript = {/files/publications/geometry/aaghhhkrv-mefpp-05.ps.gz},
htmlnote = {Also available as FSP-report S092-4, Austria, 2005, at http://www.industrial-geometry.at/techrep.php.},
abstract = {We define general Laman (count) conditions for edges and
faces of polygonal partitions in the plane. Several
well-known classes, including $k$-regular partitions,
$k$-angulations, and rank $k$ pseudo-triangulations, are
shown to fulfill such conditions. As a consequence
non-trivial perfect matchings exist between the edge sets
(or face sets) of two such structures when they live on the
same point set. We also describe a link to spanning tree
decompositions that applies to quadrangulations and certain
pseudo-triangulations.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{akpv-got-12,
author = {Oswin Aichholzer and Matias Korman and Alexander Pilz and Birgit Vogtenhuber},
title = {{{Geodesic Order Types}}},
rembooktitle = {Computing nd Combinatorics, Proc. 18$^{th}$ Annual International Computing and Combinatorics Conference},
booktitle = {{Proc. $18^{th}$ International Computing and Combinatorics Conference (COCOON 2012)}},
pages = {216--227},
year = {2012},
address = {Sydney, Australia},
month = {August},
editor = {Joachim Gudmundsson and Juli{\'a}n Mestre and Taso Viglas},
series = {Lecture Notes in Computer Science},
volume = {7434},
publisher = {Springer},
eprint = {1708.06064},
archiveprefix = {arXiv},
doi = {10.1007/978-3-642-32241-9_19},
abstract = {The geodesic between two points $a$ and $b$ in the interior of a simple polygon~$P$ is the shortest polygonal path inside $P$ that connects $a$ to $b$.
It is thus the natural generalization of straight line segments on unconstrained point sets to polygonal environments.
In this paper we use this extension to generalize the concept of the order type of a set of points in the Euclidean plane to geodesic order types.
In particular, we show that, for any set $S$ of points and an ordered subset $\blue \subseteq S$ of at least four points, one can always construct a polygon $P$ such that the points of $\blue$ define the geodesic hull of~$S$ w.r.t.~$P$, in the specified order.
Moreover, we show that an abstract order type derived from the dual of the Pappus arrangement can be realized as a geodesic order type.},
originalfile = {/geometry/cggg.bib}
}
@article{amp-ephes-13,
author = {Oswin Aichholzer and
Tillmann Miltzow and
Alexander Pilz},
title = {{{Extreme point and halving edge search in abstract order
types}}},
journal = {Comput. Geom.},
volume = {46},
number = {8},
year = {2013},
pages = {970--978},
doi = {http://dx.doi.org/10.1016/j.comgeo.2013.05.001},
bibsource = {DBLP, http://dblp.uni-trier.de},
abstract = {Many properties of finite point sets only depend on the relative position of the points, e.g., on the order type of the set.
However, many fundamental algorithms in computational geometry rely on coordinate representations.
This includes the straightforward algorithms for finding a halving line for a given planar point set, as well as finding a point on the convex hull, both in linear time.
In his monograph \emph{Axioms and Hulls}, Knuth asks whether these problems can be solved in linear time in a more abstract setting, given only the orientation of each point triple, i.e., the set's chirotope, as a source of information.
We answer this question in the affirmative.
More precisely, we can find a halving line through any given point, as well as the vertices of the convex hull edges that are intersected by the supporting line of any two given points of the set in linear time.
We first give a proof for sets realizable in the Euclidean plane and then extend the result to non-realizable abstract order types.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{amp-fdtsp-13b,
author = {Oswin Aichholzer and
Wolfgang Mulzer and
Alexander Pilz},
title = {{{Flip Distance between Triangulations of a Simple Polygon
is {NP}-Complete}}},
booktitle = {Proc. $21^{st}$ European Symposium on Algorithms (ESA 2013)},
year = {2013},
pages = {13--24},
doi = {http://dx.doi.org/10.1007/978-3-642-40450-4_2},
crossref = {DBLP:conf/esa/2013},
bibsource = {DBLP, http://dblp.uni-trier.de},
abstract = {Let $T$ be a triangulation of a simple polygon.
A \emph{flip} in~$T$ is the operation of replacing one diagonal of~$T$
by a different one such that the resulting graph is again
a triangulation. The \emph{flip distance} between two triangulations is the smallest
number of flips required to transform one triangulation into the
other. For the special case of convex polygons, the problem of determining
the shortest flip distance between two triangulations is equivalent
to determining the rotation distance between two binary trees,
a central problem which is still open after over 25 years of intensive study.
We show that computing the flip distance between two
triangulations of a simple polygon is NP-hard. This complements a recent
result that shows APX-hardness of determining the flip distance between two
triangulations of a planar point set.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{amp-fdtsp-13a,
author = {Oswin Aichholzer and
Wolfgang Mulzer and
Alexander Pilz},
title = {{{Flip Distance between Triangulations of a Simple Polygon
is {NP}-Complete}}},
booktitle = {Proc. $29^{th}$ European Workshop on Computational Geometry (EuroCG 2013)},
year = {2013},
pages = {115--118},
address = {Braunschweig, Germany},
abstract = {Let $T$ be a triangulation of a simple polygon.
A \emph{flip} in~$T$ is the operation of replacing one diagonal of~$T$
by a different one such that the resulting graph is again
a triangulation. The \emph{flip distance} between two triangulations is the smallest
number of flips required to transform one triangulation into the
other. For the special case of convex polygons, the problem of determining
the shortest flip distance between two triangulations is equivalent
to determining the rotation distance between two binary trees,
a central problem which is still open after over 25 years of intensive study.
We show that computing the flip distance between two
triangulations of a simple polygon is NP-hard. This complements a recent
result that shows APX-hardness of determining the flip distance between two
triangulations of a planar point set.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{abhpv-ltdbm-14,
author = {Aichholzer, Oswin and Barba, Luis and Hackl, Thomas
and Pilz, Alexander and Vogtenhuber, Birgit},
title = {{{Linear Transformation Distance for Bichromatic
Matchings}}},
booktitle = {Proc. 30\textsuperscript{th} Symposium on
Computational Geometry (SOCG 2014)},
remseries = {SOCG'14},
year = {2014},
isbn = {978-1-4503-2594-3},
location = {Kyoto, Japan},
pages = {154--162},
articleno = {154},
numpages = {9},
url = {http://doi.acm.org/10.1145/2582112.2582151},
doi = {http://dx.doi.org/10.1145/2582112.2582151},
acmid = {2582151},
publisher = {ACM},
remaddress = {New York, NY, USA},
keywords = {bichromatic point set, compatible matchings,
perfect matchings, reconfiguration problem,
transformation graph},
arxiv = {1312.0884v1},
pdf = {/files/publications/geometry/abhpv-ltdbm-14.pdf},
category = {3b},
thackl_label = {41C},
abstract = {Let $P=B\cup R$ be a set of $2n$ points in general
position, where $B$ is a set of $n$ blue points and
$R$ a set of $n$ red points. A \emph{$BR$-matching}
is a plane geometric perfect matching on $P$ such
that each edge has one red endpoint and one blue
endpoint. Two $BR$-matchings are compatible if their
union is also plane.\\ The \emph{transformation
graph of $BR$-matchings} contains one node for each
$BR$-matching and an edge joining two such nodes if
and only if the corresponding two $BR$-matchings are
compatible. In SoCG 2013 it has been shown by
Aloupis, Barba, Langerman, and Souvaine that this
transformation graph is always connected, but its
diameter remained an open question. In this paper we
provide an alternative proof for the connectivity of
the transformation graph and prove an upper bound of
$2n$ for its diameter, which is asymptotically
tight.},
originalfile = {/geometry/cggg.bib}
}
@book{aj-eag-13,
author = {O. Aichholzer and B. J{\"u}ttler},
title = {{{Einf{\"u}hrung in die Angewandte Geometrie}}},
publisher = {Birkh{\"a}user},
series = {Mathematik Kompakt},
htmlnote = {Verlagsseite},
abstract = {Das Buch ist an der Schnittstelle zwischen linearer
Algebra und rechnerischer Geometrie angesiedelt. Einerseits
werden die klassischen Geometrien (euklidisch, affin,
projektiv, nicht-euklidisch) mit Mitteln der linearen
Algebra behandelt. Andererseits werden grundlegende
Strukturen der rechnerischen Geometrie (Splinekurven,
Mittelachsen, Triangulierungen) und algorithmische Methoden
diskutiert. Der Schwerpunkt liegt dabei auf den
geometrischen Eigenschaften, gleichzeitig werden auch
relevante algorithmische Konzepte vorgestellt. Zahlreiche
{\"U}bungsaufgaben (mit L{\"o}sungshinweisen) erg{\"a}nzen
die Darstellung. Das Buch eignet sich f{\"u}r Studierende
aus den Fachrichtungen Mathematik, Informatik,
Maschinenbau, Bauingenieurwesen und verwandter
Studieng{\"a}nge ab dem zweiten Semester. Es kann als
Lehrbuch verwendet werden oder als erg{\"a}nzende Literatur
f{\"u}r Grundvorlesungen {\"u}ber angewandte Geometrie,
analytische Geometrie, rechnerische Geometrie
(Computational Geometry) sowie Computer Aided Geometric Design.},
year = 2013,
originalfile = {/geometry/cggg.bib}
}
@article{ak-aoten-06,
author = {O. Aichholzer and H. Krasser},
title = {{{Abstract Order Type Extension and New Results on the
Rectilinear Crossing Number}}},
year = 2006,
journal = {Computational Geometry: Theory and Applications, Special
Issue on the 21st European Workshop on Computational
Geometry},
volume = {36},
number = {1},
pages = {2--15},
category = {3a},
oaich_label = {54c},
postscript = {/files/publications/geometry/ak-aoten-06.ps.gz},
abstract = {We extend the order type data base of all realizable order
types in the plane to point sets of cardinality 11. More
precisely, we provide a complete data base of all
combinatorial different sets of up to 11 points in general
position in the plane. In addition, we develop a novel and
efficient method for a complete extension to order types of
size 12 and more in an abstract sense, that is, without the
need to store or realize the sets. The presented method is
well suited for independent computations. Thus, time
intensive investigations benefit from the possibility of
distributed computing.\\ Our approach has various
applications to combinatorial problems which are based on
sets of points in the plane. This includes classic problems
like searching for (empty) convex k-gons ('happy end problem'),
decomposing sets into convex regions, counting
structures like triangulations or pseudo-triangulations,
minimal crossing numbers, and more. We present some
improved results to several of these problems. As an
outstanding result we have been able to determine the exact
rectilinear crossing number of the complete graph $K_n$ for
up to $n=17$, the largest previous range being $n=12$, and
slightly improved the asymptotic upper bound. },
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aar-msrp-94b,
author = {O. Aichholzer and H. Alt and G. Rote},
title = {{{Matching Shapes with a Reference Point}}},
booktitle = {Proc. $10^{th}$ European Workshop on Computational
Geometry CG '94},
pages = {81--84},
year = 1994,
address = {Santander, Spain},
category = {3b},
oaich_label = {4b},
postscript = {/files/publications/geometry/aar-msrp-94.ps.gz},
abstract = {For two given point sets, we present a very simple (almost
trivial) algorithm to translate one set so that the
Hausdorff distance between the two sets is not larger than
a constant factor times the minimum Hausdorff distance
which can be achieved in this way. The algorithm just
matches the so-called Steiner points of the two sets.\\ The
focus of our paper is the general study of reference points
(like the Steiner point) and their properties with respect
to shape matching.\\ For more general transformations than
just translations, our method eliminates several degrees of
freedom from the problem and thus yields good matchings
with improved time bounds.},
originalfile = {/geometry/cggg.bib}
}
@article{aahk-tstpt-05b,
author = {O. Aichholzer and F. Aurenhammer and C. Huemer and H.
Krasser},
title = {{{Transforming Spanning Trees and Pseudo-Triangulations}}},
journal = {Information Processing Letters (IPL)},
year = 2006,
volume = {97(1)},
pages = {19--22},
category = {3a},
oaich_label = {55b},
postscript = {/files/publications/geometry/aahk-tstpt-05.ps.gz},
pdf = {/files/publications/geometry/aahk-tstpt-05.pdf},
abstract = {Let $T_{S}$ be the set of all crossing-free straight line
spanning trees of a planar $n$-point set~$S$. Consider the
graph ${\cal T}_S$ where two members $T$ and $T'$ of
$T_{S}$ are adjacent if $T$ intersects $T'$ only in points
of~$S$ or in common edges. We prove that the diameter
of~${\cal T}_S$ is $O(\log k)$, where $k$ denotes the
number of convex layers of $S$. Based on this result, we
show that the flip graph~${\cal P}_S$ of
pseudo-triangulations of~$S$ (where two
pseudo-triangulations are adjacent if they differ in
exactly one edge -- either by replacement or by removal)
has a dia\-meter of $O(n \log k)$. This sharpens a known
$O(n \log n)$ bound. Let~${\cal \widehat{P}}_S$ be the
induced subgraph of pointed pseudo-triangulations of~${\cal
P}_S$. We present an example showing that the distance
between two nodes in~${\cal \widehat{P}}_S$ is strictly
larger than the distance between the corresponding nodes
in~${\cal P}_S$. },
originalfile = {/geometry/cggg.bib}
}
@article{aackrtx-tin-96,
author = {O. Aichholzer and F. Aurenhammer and S.-W. Cheng and N.
Katoh and G. Rote and M. Taschwer and Y.-F. Xu},
title = {{{Triangulations intersect nicely}}},
journal = {Discrete \& Computational Geometry},
year = 1996,
volume = 16,
pages = {339--359},
note = {Special Issue. [SFB Report F003-030, TU Graz, Austria,
1995]},
category = {3a},
oaich_label = {7},
postscript = {/files/publications/geometry/aackrtx-tin-96.ps.gz},
abstract = {We show that there is a matching between the edges of any
two triangulations of a planar point set such that an edge
of one triangulation is matched either to the identical
edge in the other triangulation or to an edge that crosses
it. This theorem also holds for the triangles of the
triangulations and in general independence systems. As an
application, we give some lower bounds for the minimum
weight triangulation which can be computed in polynomial
time by matching and network flow techniques. We exhibit an
easy-to-recognize class of point sets for which the
minimum-weight triangulation coincides with the greedy
triangulation.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aaaj-emact-11,
author = {O. Aichholzer and W. Aigner and F. Aurenhammer and B.
J\"uttler},
title = {{{Exact medial axis computation for triangulated solids with
respect to piecewise linear metrics}}},
booktitle = {Proc. $7^{th}$ International Conference on Curves and
Surfaces 2010 (Avignon, France)},
editor = {J.D. Boissonat and M.L. Mazure and L.L. Schumaker},
address = {Avignon, France},
publisher = {Springer},
series = {Lecture Notes in Computer Science},
number = {6920},
category = {3b},
pages = {1--27},
year = {2011},
pdf = {/files/publications/geometry/aaaj-emact-11.pdf},
abstract = {We propose a novel approach for the medial axis
approximation of triangulated solids by using a polyhedral
unit ball $B$ instead of the standard Euclidean unit ball.
By this means, we compute the exact medial axis
$MA(\Omega)$ of a triangulated solid $\Omega$ with respect
to a piecewise linear (quasi-)metric $d_B$. The obtained
representation of $\Omega$ by the medial axis transform
$MAT(\Omega)$ allows for a convenient computation of the
trimmed offset of $\Omega$ with respect to $d_B$. All
calculations are performed within the field of rational
numbers, resulting in a robust and efficient implementation
of our approach. Adapting the properties of $B$ provides an
easy way to control the level of details captured by the
medial axis, making use of the implicit pruning at flat
boundary features.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aahk-tstpt-05,
author = {O. Aichholzer and F. Aurenhammer and C. Huemer and H.
Krasser},
title = {{{Transforming Spanning Trees and Pseudo-Triangulations}}},
booktitle = {Proc. $21^{th}$ European Workshop on Computational
Geometry EWCG '05},
year = 2005,
pages = {81--84},
address = {Eindhoven, The Nederlands},
category = {3b},
oaich_label = {55},
postscript = {/files/publications/geometry/aahk-tstpt-05.ps.gz},
pdf = {/files/publications/geometry/aahk-tstpt-05.pdf},
htmlnote = {Also available as FSP-report S092-10, Austria, 2006, at http://www.industrial-geometry.at/techrep.php.},
abstract = {Let $T_{S}$ be the set of all crossing-free straight line
spanning trees of a planar $n$-point set~$S$. Consider the
graph ${\cal T}_S$ where two members $T$ and $T'$ of
$T_{S}$ are adjacent if $T$ intersects $T'$ only in points
of~$S$ or in common edges. We prove that the diameter
of~${\cal T}_S$ is $O(\log k)$, where $k$ denotes the
number of convex layers of $S$. Based on this result, we
show that the flip graph~${\cal P}_S$ of
pseudo-triangulations of~$S$ (where two
pseudo-triangulations are adjacent if they differ in
exactly one edge -- either by replacement or by removal)
has a dia\-meter of $O(n \log k)$. This sharpens a known
$O(n \log n)$ bound. Let~${\cal \widehat{P}}_S$ be the
induced subgraph of pointed pseudo-triangulations of~${\cal
P}_S$. We present an example showing that the distance
between two nodes in~${\cal \widehat{P}}_S$ is strictly
larger than the distance between the corresponding nodes
in~${\cal P}_S$. },
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aabekm-nesca-00,
author = {O. Aichholzer and F. Aurenhammer and B. Brandtst{\"a}tter
and T. Ebner and H. Krasser and C. Magele},
title = {{{Niching evolution strategy with cluster algorithms}}},
booktitle = {$9^{th}$ Biennial IEEE Conf. Electromagnetic Field
Computations},
year = 2000,
address = {Milwaukee, Wisconsin, USA},
category = {7},
oaich_label = {24},
postscript = {/files/publications/geometry/aabekm-nesca-00.ps.gz},
abstract = {In most real world optimization problems one tries to
determine the global among some or even numerous local
solutions within the feasible region of parameters. On the
other hand, it could be worth to investigate some of the
local solutions as well. Therefore, a most desirable
behaviour would be, if the optimization strategy behaves
globally and yields additional information about local
minima detected on the way to the global solution. In this
paper a clustering algorithm has been implemented into an
Higher Order Evolution Strategy in order to achieve these
goals.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{a-lpt-95,
author = {O. Aichholzer},
title = {{{Local properties of triangulations}}},
booktitle = {Proc. $11^{th}$ European Workshop on Computational
Geometry CG '95},
pages = {27--30},
year = 1995,
address = {Hagenberg/Linz, Austria},
category = {3b},
oaich_label = {8},
postscript = {/files/publications/geometry/a-lpt-95.ps.gz},
abstract = {In this paper we study local properties of two well known
triangulations of a planar point set $S$, both of which are
defined in a non-local way. The first one is the greedy
triangulation (GT) that is defined procedurally: it can be
obtained by starting with the empty set and at each step
adding the shortest compatible edge between two points of
$S$, where a compatible edge is defined to be an edge that
does not cross any of the previously inserted edges. The
other triangulation we deal with is the minimum-weight
triangulation (MWT) which minimizes the sum of the length
of the edges among all possible triangulations of $S$. We
present several results on exclusion- and inclusion-regions
for these two triangulations.},
originalfile = {/geometry/cggg.bib}
}
@article{abdgh-cgm-08c,
author = {O. Aichholzer and S. Bereg and A. Dumitrescu and A.
Garc\'{\i}a and C. Huemer and F. Hurtado and M. Kano and A.
M{\'a}rquez and D. Rappaport and S. Smorodinsky and D.
Souvaine and J. Urrutia and D. Wood},
title = {{{Compatible Geometric Matchings}}},
year = 2008,
volume = {31},
number = {},
journal = {Electronic Notes in Discrete Mathematics},
pages = {201--206},
category = {3a},
oaich_label = {76a},
doi = {http://dx.doi.org/10.1016/j.endm.2008.06.040},
pdf = {/files/publications/geometry/abdgh-cgm-08.pdf},
abstract = {Abstract: This paper studies non-crossing geometric
perfect matchings. Two such perfect matchings are
compatible if they have the same vertex set and their union
is also non-crossing. Our first result states that for any
two perfect matchings $M$ and $M'$ of the same set of $n$
points, for some $k \in O(log n)$, there is a sequence of
perfect matchings $M = M_0,M_1, . . . ,M_k = M'$, such that
each $M_i$ is compatible with $M_{i+1}$. This improves the
previous best bound of $k \leq n-2$. We then study the
conjecture: every perfect matching with an even number of
edges has an edge-disjoint compatible perfect matching. We
introduce a sequence of stronger conjectures that imply
this conjecture, and prove the strongest of these
conjectures in the case of perfec matchings that consist of
vertical and horizontal segments. Finally, we prove that
every perfect matching with $n$ edges has an edge-disjoint
compatible matching with approximately $4n/5$ edges. },
originalfile = {/geometry/cggg.bib}
}
@article{aaiklr-vsac-00,
author = {O. Aichholzer and F. Aurenhammer and C. Icking and R.
Klein and E. Langetepe and G. Rote},
title = {{{Generalized self-approaching curves}}},
journal = {Discrete Applied Mathematics},
year = 2001,
note = {Special Issue. [SFB-Report F003-134, TU Graz, Austria,
1998]},
pages = {3--24},
volume = {109},
number = {1-2},
category = {3a},
oaich_label = {20b},
pdf = {/files/publications/geometry/aaiklr-gsac-01.pdf},
postscript = {/files/publications/geometry/aaiklr-vsac-00.ps.gz},
abstract = {We consider all planar oriented curves that have the
following property depending on a fixed angle $\varphi$.
For each point $B$ on the curve, the rest of the curve lies
inside a wedge of angle $\varphi$ with apex in $B$. This
property restrains the curve's meandering, and for $\varphi
\leq \Pi/2$ this means that a point running along the curve
always gets closer to all points on the remaining part. For
all $\varphi < \Pi$, we provide an upper bound $c(\varphi)$
for the length of such a curve, divided by the distance
between its endpoints, and prove this bound to be tight. A
main step is in proving that the curve's length cannot
exceed the perimeter of its convex hull, divided by
$1+\cos(\varphi)$.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ak-psotd-01,
author = {O. Aichholzer and H. Krasser},
title = {{{The Point Set Order Type Data Base: A Collection of
Applications and Results}}},
booktitle = {Proc. $13th$ Annual Canadian Conference on Computational
Geometry CCCG 2001},
pages = {17--20},
year = 2001,
address = {Waterloo, Ontario, Canada},
category = {3b},
oaich_label = {34},
postscript = {/files/publications/geometry/ak-psotd-01.ps.gz},
htmlnote = {See also our order
type homepage.},
abstract = {Order types are a common tool to provide the combinatorial
structure of point sets in the plane. For many problems in
combinatorial and computational geometry only the order
type of the underlying point set has to be considered.
Recently a complete order type data base of $n$-point sets
has been developed for $n\leq 10$, which gives a way to
examine the combinatorial properties of all possible point
sets for fixed size $n$. Based on this result we present
applications and results for problems concerning
intersection properties, convexity, crossing-free straight
line graphs, and others, thus confirming or disproving
several conjectures on these topics. Besides providing
concrete results the aim of this work is to stimulate
further research by revealing structural relations of
extreme examples for $17$ geometrical and combinatorial
problems.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ahst-dbcpt-03,
author = {O. Aichholzer and M. Hoffmann and B. Speckmann and C. D.
T\'oth},
title = {{{Degree Bounds for Constrained Pseudo-Triangulations}}},
booktitle = {Proc. $15th$ Annual Canadian Conference on Computational
Geometry CCCG 2003},
pages = {155--158},
year = 2003,
address = {Halifax, Nova Scotia, Canada},
category = {3b},
oaich_label = {48},
postscript = {/files/publications/geometry/ahst-dbcpt-03.ps.gz},
abstract = {We introduce the concept of a constrained pointed
pseudo-triangulation $\mathcal{T}_G$ of a point set $S$
with respect to a pointed planar straight line graph $G =
(S, E)$. For the case that $G$ forms a simple polygon $P$
with vertex set $S$ we give tight bounds on the vertex
degree of $\mathcal{T}_G$. },
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aoss-nptcp-03,
author = {O. Aichholzer and D. Orden and F. Santos and B.
Speckmann},
title = {{{On the Number of Pseudo-Triangulations of Certain Point
Sets}}},
booktitle = {Proc. $15th$ Annual Canadian Conference on Computational
Geometry CCCG 2003},
pages = {141--144},
year = 2003,
address = {Halifax, Nova Scotia, Canada},
category = {3b},
oaich_label = {49},
postscript = {/files/publications/geometry/aoss-nptcp-03.ps.gz},
abstract = {We compute the exact number of pseudo-triangulations for
two prominent point sets, namely the so-called double
circle and the double chain. We also derive a new
asymptotic lower bound for the maximal number of
pseudo-triangulations which lies significantly above the
related bound for triangulations. },
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aaadjr-twca-11,
author = {O.~Aichholzer and W.~Aigner and F.~Aurenhammer and
K.\v{C}~Dobi\'a\v{s}ov\'a, B.~J\"uttler and G.~Rote},
title = {{{Triangulations with circular arcs}}},
booktitle = {$19^{th}$ Symposium on Graph Drawing 2011 (Eindhoven, The
Netherlands)},
category = {3b},
pages = {296--307},
year = 2011,
pdf = {/files/publications/geometry/aaadjr-twca-11.pdf},
abstract = {An important objective in the choice of a triangulation of
a given point set is that the smallest angle becomes as
large as possible. In the straight line case, it is known
that the Delaunay triangulation is optimal in this
respect.We propose and study the concept of a circular arc
triangulation, a simple and effective alternative that
offers flexibility for additionally enlarging small
angles.We show that angle optimization and related
questions lead to linear programming problems that can be
formulated as simple graph-theoretic problems, and we
define flipping operations in arc triangles. Moreover,
special classes of arc triangulations are considered, for
applications in graph drawing and finite element methods.},
originalfile = {/geometry/cggg.bib}
}
@article{aaadjr-twca-15,
author = {O.~Aichholzer and W.~Aigner and F.~Aurenhammer and
K.\v{C}~Dobi\'a\v{s}ov\'a, B.~J\"uttler and G.~Rote},
title = {{{Triangulations with circular arcs}}},
category = {3a},
journal = {Journal of Graph Algorithms and Applications},
year = {2015},
volume = {19},
number = {1},
pages = {43--65},
doi = {10.7155/jgaa.00346},
pdf = {/files/publications/geometry/aaadjr_tca_15.pdf},
abstract = {An important objective in the choice of a triangulation of
a given point set is that the smallest angle becomes as
large as possible. In the straight line case, it is known
that the Delaunay triangulation is optimal in this
respect.We propose and study the concept of a circular arc
triangulation, a simple and effective alternative that
offers flexibility for additionally enlarging small
angles.We show that angle optimization and related
questions lead to linear programming problems that can be
formulated as simple graph-theoretic problems, and we
define flipping operations in arc triangles. Moreover,
special classes of arc triangulations are considered, for
applications in finite element methods and graph drawing.},
originalfile = {/geometry/cggg.bib}
}
@article{aabk-ptsnt-03,
author = {O. Aichholzer and F. Aurenhammer and P. Brass and H.
Krasser},
title = {{{Pseudo-Triangulations from Surfaces and a Novel Type of
Edge Flip}}},
journal = {SIAM Journal on Computing},
volume = {32},
year = {2003},
pages = {1621--1653},
abstract = {We prove that planar pseudo-triangulations have
realizations as polyhedral surfaces in three-space. Two
main implications are presented: The spatial embedding
leads to a novel flip operation that allows for a drastical
reduction of flip distances, especially between (full)
triangulations. Moreover, several key results for
triangulations, like flipping to optimality, (constrained)
Delaunayhood, and a convex polytope representation, are
extended to pseudo-triangulations in a natural way.},
address = {Graz, Austria},
category = {3a},
oaich_label = {41},
postscript = {/files/publications/geometry/aabk-ptsnt-03.ps.gz},
originalfile = {/geometry/cggg.bib}
}
@article{arsv-pdpg-11,
author = {O. Aichholzer and G. Rote and A. Schulz and B.
Vogtenhuber},
title = {{{Pointed Drawings of Planar Graphs}}},
year = 2012,
journal = {Computational Geometry: Theory and Applications},
pages = {482--494},
note = {special issue of CCCG 2007},
category = {3a},
doi = {10.1016/j.comgeo.2010.08.001},
pdf = {/files/publications/geometry/arsv-pdpg-11.pdf},
abstract = {We study the problem how to draw a planar graph such that
every vertex is incident to an angle greater than $\pi$. In
general a straight-line embedding cannot guarantee this
property. We present algorithms which construct such
drawings with either tangent-continuous biarcs or quadratic
B\'ezier curves (parabolic arcs), even if the
\mbox{positions} of the vertices are predefined by a given
plane straight-line embedding of the graph. Moreover, the
graph can be embedded with circular arcs if the vertices
can be placed arbitrarily. The topic is related to
non-crossing drawings of multigraphs and vertex labeling.},
originalfile = {/geometry/cggg.bib}
}
@article{aarx-clgta-99,
author = {O. Aichholzer and F. Aurenhammer and G. Rote and Y.-F.
Xu},
title = {{{Constant-level greedy triangulations approximate the {MWT}
well}}},
journal = {Journal of Combinatorial Optimization},
year = 1998,
volume = 2,
pages = {361--369},
category = {3a},
oaich_label = {12},
note = {[SFB-Report F003-050, TU Graz, Austria, 1995]},
postscript = {/files/publications/geometry/aarx-clgta-99.ps.gz},
abstract = {The well-known greedy triangulation $GT(S)$ of a finite
point set $S$ is obtained by inserting compatible edges in
increasing length order, where an edge is compatible if it
does not cross previously inserted ones. Exploiting the
concept of so-called light edges, we introduce a definition
of $GT(S)$ that does not rely on the length ordering of the
edges. Rather, it provides a decomposition of $GT(S)$ into
levels, and the number of levels allows us to bound the
total edge length of $GT(S)$. In particular, we show
$|GT(S)| \leq 3 \cdot 2^{k+1} |MWT(S)|$, where $k$ is the
number of levels and $MWT(S)$ is the minimum-weight
triangulation of $S$.},
originalfile = {/geometry/cggg.bib}
}
@article{abdhkkrsu-gt-05,
author = {O. Aichholzer and D. Bremner and E.D. Demaine and F.
Hurtado and E. Kranakis and H. Krasser and S. Ramaswami and
S. Sethia and J. Urrutia},
title = {{{Games on Triangulations}}},
journal = {Theoretical Computer Science},
year = 2005,
volume = {343},
number = {1-2},
pages = {42-71},
category = {3a},
oaich_label = {59},
pdf = {/files/publications/geometry/abdhkkrsu-gt-05.pdf},
abstract = {We analyze several perfect-information combinatorial games
played on planar triangulations. We describe main broad
categories of these games and provide in various situations
polynomial-time algorithms to determine who wins a given
game under optimal play, and ideally, to find a winning
strategy. Relations to relevant existing combinatorial
games, such as Kayles, are also shown.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aadddhlrssw-cpwlv-11,
author = {Oswin Aichholzer and Greg Aloupis and Erik D. Demaine and
Martin L. Demaine and Vida Dujmovi\'c and Ferran Hurtado
and Anna Lubiw and G\"unter Rote and Andr\'e Schulz and
Diane L. Souvaine and Andrew Winslow},
title = {{{Convexifying Polygons Without Losing Visibilities}}},
booktitle = {Proc. $23^{rd}$ Annual Canadian Conference on
Computational Geometry CCCG 2011},
pages = {229--234},
year = 2011,
address = {Toronto, Canada},
category = {3b},
pdf = {/files/publications/geometry/aadddhlrssw-cpwlv-11.pdf},
abstract = {We show that any simple $n$-vertex polygon can be made
convex, without losing internal visibilities between
vertices, using $n$ moves. Each move translates a vertex of
the current polygon along an edge to a neighbouring vertex.
In general, a vertex of the current polygon represents a
set of vertices of the original polygon that have become
co-incident. We also show how to modify the method so that
vertices become very close but not co-incident. The proof
involves a new visibility property of polygons, namely that
every simple polygon has a visibility-increasing edge
where, as a point travels from one endpoint of the edge to
the other, the visibility region of the point increases.},
originalfile = {/geometry/cggg.bib}
}
@article{agor-nlbnk-06,
author = {O. Aichholzer and J. Garc\'{\i}a and D. Orden and P.A.
Ramos},
title = {New lower bounds for the number of~$(\leq k)$-edges and
the rectilinear crossing number of~$K_n$},
journal = {Discrete \& Computational Geometry},
year = 2007,
volume = {38},
category = {3a},
oaich_label = {64},
pages = {1--14},
postscript = {/files/publications/geometry/agor-nlbnk-06.ps.gz},
htmlnote = {Also available as FSP-report S092-20, Austria, 2006, at http://www.industrial-geometry.at/techrep.php.},
abstract = {We provide a new lower bound on the number of~$(\leq
k)$-edges on a set of~$n$ points in the plane in general
position. We show that for $0 \leq k \leq
\lfloor\frac{n-2}{2}\rfloor$ the number of~$(\leq k)$-edges
is at least $ E_k(S) \geq 3 {k+2 \choose 2} +
\sum_{j=\lfloor\frac{n}{3}\rfloor}^k (3j-n+3)$, which, for
$k\geq \lfloor \frac{n}{3}\rfloor$, improves the previous
best lower bound.\\ As a main consequence, we obtain a new
lower bound on the rectilinear crossing number of the
complete graph or, in other words, on the minimum number of
convex quadrilaterals determined by~$n$ points in the plane
in general position. We show that the crossing number is at
least $ \left(\frac{41}{108}+\varepsilon \right) {n \choose
4} + O(n^3) \geq 0.379631 {n \choose 4} + O(n^3)$, which
improves the previous bound of~$0.37533 {n \choose 4} +
O(n^3)$ and approaches the best known upper bound $0.38058
{n \choose 4}$.\\ The proof is based on a result about the
structure of sets attaining the rectilinear crossing
number, for which we show that the convex hull is always a
triangle.\\ Further implications include improved results
for small values of $n$. We extend the range of known
values for the rectilinear crossing number, namely by
$cr(K_{19})=1318$ and $cr(K_{21})=2055$. Moreover we
provide improved upper bounds on the maximum number $h_n$
of halving edges a point set can have.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aoss-nptcp-04,
author = {O. Aichholzer and D. Orden and F. Santos and B.
Speckmann},
title = {{{On the Number of Pseudo-Triangulations of Certain Point
Sets}}},
booktitle = {Proc. $20^{th}$ European Workshop on Computational
Geometry EWCG '04},
year = 2004,
pages = {119--122},
address = {Sevilla, Spain},
category = {3b},
oaich_label = {49b},
postscript = {/files/publications/geometry/aoss-nptcp-04.ps.gz},
abstract = {We compute the exact number of pseudo-triangulations for
two prominent point sets, namely the so-called double
circle and the double chain. We also derive a new
asymptotic lower bound for the maximal number of
pseudo-triangulations which lies significantly above the
related bound for triangulations. },
originalfile = {/geometry/cggg.bib}
}
@inproceedings{a-pt-99,
author = {O. Aichholzer},
title = {{{The Path of a Triangulation}}},
booktitle = {Proc. $15^{th}$ Ann. ACM Symp. Computational Geometry},
pages = {14--23},
year = 1999,
address = {Miami Beach, Florida, USA},
category = {3b},
oaich_label = {17a},
postscript = {/files/publications/geometry/a-pt-99.ps.gz},
htmlnote = {For an implementation see my page on triangulation
counting.},
abstract = {For a planar point set $S$ let $T$ be a triangulation of
$S$ and $l$ a line properly intersecting $T$. We show that
there always exists a unique path in $T$ with certain
properties with respect to $l$. This path is then
generalized to (non triangulated) point sets restricted to
the interior of simple polygons. This so-called
triangulation path enables us to treat several
triangulation problems on planar point sets in a divide \&
conquer-like manner. For example, we give the first
algorithm for counting triangulations of a planar point set
which is observed to run in time sublinear in the number of
triangulations. Moreover, the triangulation path proves to
be useful for the computation of optimal triangulations.},
originalfile = {/geometry/cggg.bib}
}
@article{aa-chh-96,
author = {O. Aichholzer and F. Aurenhammer},
title = {{{Classifying hyperplanes in hypercubes}}},
journal = {SIAM Journal on Discrete Mathematics},
year = 1996,
volume = 9,
number = 2,
pages = {225--232},
category = {3a},
oaich_label = {3},
note = {[IIG-Report-Series 408, TU Graz, Austria, 1995]},
postscript = {/files/publications/geometry/aa-chh-96.ps.gz},
abstract = {We consider hyperplanes spanned by vertices of the unit
$d$-cube. We classify these hyperplanes by parallelism to
coordinate axes, by symmetry of the $d$-cube vertices they
avoid, as well as by so-called hull-honesty. (Hull-honest
hyperplanes are those whose intersection figure with the
$d$-cube coincides with the convex hull of the $d$-cube
vertices they contain; they do not cut $d$-cube edges
properly.) We describe relationships between these classes,
and give the exact number of hull-honest hyperplanes, in
general dimensions. An experimental enumeration of all
spanned hyperplanes up to dimension eight showed us the
intrinsic difficulty of developing a general enumeration
scheme. Motivation for considering such hyperplanes stems
from coding theory, from linear programming, and from the
theory of machine learning.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{auv-b6bps-13,
author = {O.~Aichholzer and J.~Urrutia and B.~Vogtenhuber},
title = {{{Balanced 6-holes in bichromatic point sets}}},
booktitle = {Proc. of the $16^{th}$ Japan Conference on Discrete and
Computational Geometry and Graphs (JCDCG$^2$ 2013)},
year = 2013,
address = {Tokyo, Japan},
category = {3b},
pdf = {/files/publications/geometry/auv-b6bps-13.pdf},
abstract = {We consider an Erd{\H{o}}s type question on $k$-holes (empty
$k$-gons) in bichromatic point sets. For a bichromatic
point set $S = R \cup B$, a balanced $2k$-hole in $S$ is
spanned by $k$ points of $R$ and $k$ points of $B$. We show
that if $|R| = |B| = n$, then the number of balanced
6-holes in $S$ is at least $1/45n^2-\Theta(n)$.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aaiklr-gsac-98a,
author = {O. Aichholzer and F. Aurenhammer and C. Icking and R.
Klein and E. Langetepe and G. Rote},
title = {{{Generalized self-approaching curves}}},
booktitle = {Proc. $14^{th}$ European Workshop on Computational
Geometry CG '98},
pages = {15--18},
year = 1998,
address = {Barcelona, Spain},
category = {3b},
oaich_label = {20},
postscript = {/files/publications/geometry/aaiklr-gsac-98a.ps.gz},
abstract = {We consider all planar oriented curves that have the
following property. For each point $B$ on the curve, the
rest of the curve lies inside a wedge of angle $\varphi$
with apex in $B$, where $\varphi < \Pi$ is fixed. This
property restrains the curve's meandering. we provide an
upper bound $c(\varphi)$ for the length of such a curve,
divided by the distance between its endpoints, and prove
this bound to be tight. A main step is in proving that the
curve's length cannot exceed the perimeter of its convex
hull, divided by $1+\cos(\varphi)$.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{arsv-pdpg-07,
author = {O. Aichholzer and G. Rote and A. Schulz and B.
Vogtenhuber},
title = {{{Pointed Drawings of Planar Graphs}}},
booktitle = {Proc. $19th$ Annual Canadian Conference on Computational
Geometry CCCG 2007},
pages = {237--240},
year = 2007,
address = {Ottawa, Ontario, Canada},
category = {3b},
oaich_label = {72},
pdf = {/files/publications/geometry/arsv-pdpg-07.pdf},
abstract = {We study the problem how to draw a planar graph such that
every vertex is incident to an angle greater than $\pi$. In
general a straight-line embedding cannot guarantee this
property. We present algorithms which construct such
drawings with either tangent-continuous biarcs or quadratic
B\'ezier curves (parabolic arcs), even if the
\mbox{positions} of the vertices are predefined by a given
plane straight-line embedding of the graph. Moreover, the
graph can be embedded with circular arcs if the vertices
can be placed arbitrarily. The topic is related to
non-crossing drawings of multigraphs and vertex labeling.},
originalfile = {/geometry/cggg.bib}
}
@article{aaw-afa-02,
author = {O. Aichholzer and F. Aurenhammer and T. Werner},
title = {{{Algorithmic Fun - {A}balone}}},
journal = {Special Issue on Foundations of Information Processing of
{TELEMATIK}},
pages = {4--6},
year = {2002},
volume = 1,
address = {Graz, Austria},
category = {4a},
oaich_label = {38},
postscript = {/files/publications/geometry/aaw-afa-02.ps.gz},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ahrst-pcdpc-05,
author = {O. Aichholzer and C. Huemer and S. Renkl and B. Speckmann
and C. D. T\'oth},
title = {{{On Pseudo-Convex Decompositions, Partitions, and
Coverings}}},
booktitle = {Proc. $21^{th}$ European Workshop on Computational
Geometry EWCG '05},
year = 2005,
pages = {89--92},
address = {Eindhoven, The Nederlands},
category = {3b},
oaich_label = {56},
postscript = {/files/publications/geometry/ahrst-pcdpc-05.ps.gz},
pdf = {/files/publications/geometry/ahrst-pcdpc-05.pdf},
htmlnote = {Also available as FSP-report S092-3, Austria, 2005, at http://www.industrial-geometry.at/techrep.php.},
abstract = {We introduce pseudo-convex decompositions, partitions, and
coverings for planar point sets. They are natural
extensions of their convex counterparts that use both
convex polygons and pseudo-triangles. We discuss some of
their basic combinatorial properties and establish upper
and lower bounds on their complexity.},
originalfile = {/geometry/cggg.bib}
}
@article{aap-qpssc-03,
author = {O. Aichholzer and F. Aurenhammer and B. Palop},
title = {{{Quickest Paths, Straight Skeletons, and the City {V}oronoi
Diagram}}},
journal = {Discrete \& Computational Geometry},
volume = 31,
number = 1,
pages = {17--35},
year = {2004},
category = {3a},
oaich_label = {37a},
postscript = {/files/publications/geometry/aap-qpssc-03.ps.gz},
abstract = {The city Voronoi diagram is induced by quickest paths, in
the $L_1$~plane speeded up by an isothetic transportation
network. We investigate the rich geometric and algorithmic
properties of city Voronoi diagrams, and report on their
use in processing quickest-path queries.\\ In doing so, we
revisit the fact that not every Voronoi-type diagram has
interpretations in both the distance model and the
wavefront model. Especially, straight skeletons are a
relevant example where an interpretation in the former
model is lacking. We clarify the relation between these
models, and further draw a connection to the
bisector-defined abstract Voronoi diagram model, with the
particular goal of computing the city Voronoi diagram
efficiently. },
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ar-qdbsb-04,
author = {O. Aichholzer and K. Reinhardt},
title = {{{A quadratic distance bound on sliding between
crossing-free spanning trees - Extended Abstract}}},
booktitle = {Proc. $20^{th}$ European Workshop on Computational
Geometry EWCG '04},
year = 2004,
pages = {13--16},
address = {Sevilla, Spain},
category = {3b},
oaich_label = {51},
postscript = {/files/publications/geometry/ar-qdbsb-04.ps.gz},
abstract = {Let $S$ be a set of $n$ points in the plane and let
${\mathcal T}_S$ be the set of all crossing-free spanning
trees of $S$. We show that any two trees in ${\mathcal
T}_S$ can be transformed into each other by $O(n^2)$ local
and constant-size edge slide operations. No polynomial
upper bound for this task has been known, but in~\cite{AAH}
a bound of $O(n^2 \log n)$ operations was conjectured.},
originalfile = {/geometry/cggg.bib}
}
@article{abdgh-cgm-08b,
author = {O. Aichholzer and S. Bereg and A. Dumitrescu and A.
Garc\'{\i}a and C. Huemer and F. Hurtado and M. Kano and A.
M{\'a}rquez and D. Rappaport and S. Smorodinsky and D.
Souvaine and J. Urrutia and D. Wood},
title = {{{Compatible Geometric Matchings}}},
year = 2009,
volume = {42},
number = {6-7},
journal = {Computational Geometry: Theory and Applications},
pages = {617--626},
category = {3a},
oaich_label = {76a},
pdf = {/files/publications/geometry/abdgh-cgm-08.pdf},
abstract = {Abstract: This paper studies non-crossing geometric
perfect matchings. Two such perfect matchings are
compatible if they have the same vertex set and their union
is also non-crossing. Our first result states that for any
two perfect matchings $M$ and $M'$ of the same set of $n$
points, for some $k \in O(log n)$, there is a sequence of
perfect matchings $M = M_0,M_1, . . . ,M_k = M'$, such that
each $M_i$ is compatible with $M_{i+1}$. This improves the
previous best bound of $k \leq n-2$. We then study the
conjecture: every perfect matching with an even number of
edges has an edge-disjoint compatible perfect matching. We
introduce a sequence of stronger conjectures that imply
this conjecture, and prove the strongest of these
conjectures in the case of perfec matchings that consist of
vertical and horizontal segments. Finally, we prove that
every perfect matching with $n$ edges has an edge-disjoint
compatible matching with approximately $4n/5$ edges. },
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aaiklr-gsac-98b,
author = {O. Aichholzer and F. Aurenhammer and C. Icking and R.
Klein and E. Langetepe and G. Rote},
title = {{{Generalized self-approaching curves}}},
booktitle = {Proc. $9^{th}$ Int. Symp. Algorithms and Computation
ISAAC'98, Lecture Notes in Computer Science},
pages = {317--326},
year = 1998,
volume = 1533,
address = {Taejon, Korea},
publisher = {Springer Verlag},
category = {3b},
oaich_label = {20a},
postscript = {/files/publications/geometry/aaiklr-gsac-98b.ps.gz},
abstract = {We consider all planar oriented curves that have the
following property depending on a fixed angle $\varphi$.
For each point $B$ on the curve, the rest of the curve lies
inside a wedge of angle $\varphi$ with apex in $B$. This
property restrains the curve's meandering, and for $\varphi
\leq \Pi/2$ this means that a point running along the curve
always gets closer to all points on the remaining part. For
all $\varphi < \Pi$, we provide an upper bound $c(\varphi)$
for the length of such a curve, divided by the distance
between its endpoints, and prove this bound to be tight. A
main step is in proving that the curve's length cannot
exceed the perimeter of its convex hull, divided by
$1+\cos(\varphi)$.},
originalfile = {/geometry/cggg.bib}
}
@techreport{aar-ogosa-95,
author = {O. Aichholzer and F. Aurenhammer and G. Rote},
title = {{{Optimal graph orientation with storage applications}}},
institution = {SFB 'Optimierung und Kontrolle', TU Graz, Austria},
year = 1995,
type = {SFB-Report},
number = {F003-51},
category = {5},
oaich_label = {13},
postscript = {/files/publications/geometry/aar-ogosa-95.ps.gz},
abstract = {We show that the edges of a graph with maximum edge
density $d$ can always be oriented such that each vertex
has in-degree at most $d$. Hence, for arbitrary graphs,
edges can always be assigned to incident vertices as
uniformly as possible. For example, in-degree 3 is achieved
for planar graphs. This immediately gives a space-optimal
data structure that answers edge membership queries in a
maximum edge density-$d$ graph in $O(\log d)$ time.},
originalfile = {/geometry/cggg.bib}
}
@article{abdmss-lpuof-02,
author = {O. Aichholzer and D. Bremner and E.D. Demaine and D.
Meijer and V. Sacrist\'{a}n and M. Soss},
title = {{{Long Proteins with Unique Optimal Foldings in the H-P
Model}}},
journal = {Computational Geometry: Theory and Applications},
year = 2003,
pages = {139--159},
volume = {25},
category = {3a},
oaich_label = {30a},
postscript = {/files/publications/geometry/abdmss-lpuof-02.ps.gz},
abstract = {It is widely accepted that (1) the natural or folded state
of proteins is a global energy minimum, and (2) in most
cases proteins fold to a unique state determined by their
amino acid sequence. The H-P (hydrophobic-hydrophilic)
model is a simple combinatorial model designed to answer
qualitative questions about the protein folding process. In
this paper we consider a problem suggested by Brian Hayes
in 1998: what proteins in the two-dimensional H-P model
have \emph{unique} optimal (minimum energy) foldings? In
particular, we prove that there are closed chains of
monomers (amino acids) with this property for all (even)
lengths; and that there are open monomer chains with this
property for all lengths divisible by four.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{abdhkkrsu-pwt-02,
author = {O. Aichholzer and D. Bremner and E.D. Demaine and F.
Hurtado and E. Kranakis and H. Krasser and S. Ramaswami and
S. Sethia and J. Urrutia},
title = {{{Playing with Triangulations}}},
booktitle = {Proc. Japan Conference on Discrete and Computational
Geometry JCDCG 2002},
pages = {46--54},
year = 2002,
address = {Tokyo, Japan},
category = {3b},
oaich_label = {43},
postscript = {/files/publications/geometry/abdhkkrsu-pwt-02.ps.gz},
abstract = {We analyze several perfect-information combinatorial games
played on planar triangulations. We describe main broad
categories of these games and provide in various situations
polynomial-time algorithms to determine who wins a given
game under optimal play, and ideally, to find a winning
strategy. Relations to relevant existing combinatorial
games, such as Kayles, are also shown.},
originalfile = {/geometry/cggg.bib}
}
@article{aar-msrp-97,
author = {O. Aichholzer and H. Alt and G. Rote},
title = {{{Matching Shapes with a Reference Point}}},
journal = {Int'l Journal of Computational Geometry \& Applications},
year = 1997,
volume = 7,
number = 4,
pages = {349--363},
category = {3a},
oaich_label = {4a},
postscript = {/files/publications/geometry/aar-msrp-97.ps.gz},
abstract = {For two given point sets, we present a very simple (almost
trivial) algorithm to translate one set so that the
Hausdorff distance between the two sets is not larger than
a constant factor times the minimum Hausdorff distance
which can be achieved in this way. The algorithm just
matches the so-called Steiner points of the two sets.\\ The
focus of our paper is the general study of reference points
(like the Steiner point) and their properties with respect
to shape matching.\\ For more general transformations than
just translations, our method eliminates several degrees of
freedom from the problem and thus yields good matchings
with improved time bounds.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ahrst-pcdpc-06,
author = {O. Aichholzer and C. Huemer and S. Renkl and B. Speckmann
and C. D. T\'oth},
title = {{{Decompositions, Partitions, and Coverings with Convex
Polygons and Pseudo-Triangles}}},
booktitle = {Proceedings $31^{st}$ International Symposium on
Mathematical Foundations of Computer Science, Lecture Notes
in Computer Science},
editor = {Rastislav Kr{\'a}lovic,Pawel Urzyczyn},
year = 2006,
volume = {4162},
pages = {86-97},
address = {Star{\'a} Lesn{\'a}, Slovakia},
category = {3b},
oaich_label = {56b},
pdf = {/files/publications/geometry/ahrst-pcdpc-06.pdf},
abstract = {We propose a novel subdivision of the plane that consists
of both convex polygons and pseudo-triangles. This
pseudo-convex decomposition is significantly sparser than
either convex decompositions or pseudo-triangulations for
planar point sets and simple polygons. We also introduce
pseudo-convex partitions and coverings. We establish some
basic properties and give combinatorial bounds on their
complexity. Our upper bounds depend on new Ramsey-type
results concerning disjoint empty convex k-gons in point
sets.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{arss-zppt-03,
author = {O. Aichholzer and G. Rote and B. Speckmann and I.
Streinu},
title = {{{The Zigzag Path of a Pseudo-Triangulation}}},
booktitle = {Lecture Notes in Computer Science, Proc. 8th International
Workshop on Algorithms and Data Structures (WADS)},
volume = {2748},
pages = {377--389},
year = 2003,
category = {3b},
oaich_label = {44},
abstract = {We define the zigzag path of a pseudo-triangulation, a
concept generalizing the path of a triangulation of a point
set. The pseudo-tri\-an\-gu\-la\-tion zigzag path allows us
to use divide-and-conquer type of approaches for suitable
(i.e., decomposable) problems on
pseudo-tri\-an\-gu\-la\-tions. For this we provide an
algorithm that enumerates all pseudo-triangulation zigzag
paths (of all pseudo-triangulations of a given point set
with respect to a given line) in $O(n^2)$ time per path and
$O(n^2)$ space, where $n$ is the number of points. We
illustrate applications of our scheme which include a novel
algorithm to count the number of pseudo-triangulations of a
point set. },
postscript = {/files/publications/geometry/arss-zppt-03.ps.gz},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aak-aptnl-03,
author = {O. Aichholzer and F. Aurenhammer and H. Krasser},
title = {{{Adapting (Pseudo)-Triangulations with a Near-Linear Number
of Edge Flips}}},
booktitle = {Lecture Notes in Computer Science 2748, Proc. 8th
International Workshop on Algorithms and Data Structures
(WADS)},
volume = {2748},
pages = {12--24},
year = 2003,
category = {3b},
oaich_label = {46},
abstract = {We provide two results on flip distances in
pseudo-triangulations -- for minimum pseudo-triangulations
when using traditional flips operations, as well as for
triangulations when a novel and natural edge flip operation
is included into the repertoire of admissible flips. The
obtained flip distance lengths are $O(n \log^2 n)$ and $O(n
\log n)$, respectively. Our results partially rely on new
partitioning results for pseudo-triangulations which may be
of separate interest.},
postscript = {/files/publications/geometry/aak-aptnl-03.ps.gz},
originalfile = {/geometry/cggg.bib}
}
@article{ahkst-pcdpc-07,
author = {O. Aichholzer and C. Huemer and S. Kappes and B. Speckmann
and C. D. T\'oth},
title = {{{Decompositions, Partitions, and Coverings with Convex
Polygons and Pseudo-Triangles}}},
journal = {Graphs and Combinatorics},
year = 2007,
volume = {23(5)},
pages = {481-507},
category = {3a},
oaich_label = {56c},
pdf = {/files/publications/geometry/ahkst-pcdpc-07.pdf},
abstract = {We propose a novel subdivision of the plane that consists
of both convex polygons and pseudo-triangles. This
pseudo-convex decomposition is significantly sparser than
either convex decompositions or pseudo-triangulations for
planar point sets and simple polygons. We also introduce
pseudo-convex partitions and coverings. We establish some
basic properties and give combinatorial bounds on their
complexity. Our upper bounds depend on new Ramsey-type
results concerning disjoint empty convex k-gons in point
sets.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ass-ppt-02,
author = {O. Aichholzer and B. Speckmann and I. Streinu},
title = {{{The Path of a Pseudo-Triangulation}}},
booktitle = {Abstracts of the DIMACS Workshop on Computational Geometry
2002},
pages = {2},
year = 2002,
address = {Piscataway (NJ), USA},
category = {3b},
oaich_label = {44b},
postscript = {/files/publications/geometry/ass-ppt-02.pdf},
abstract = {We define the path of a pseudo-triangulation, a data
structure generalizing the path of a triangulation of a
point set. This structure allows us to use
divide-and-conquer type of approaches for suitable (i.e.
decomposable) problems on pseudo-triangulations. We
illustrate this method by presenting a novel algorithm that
counts the number of pseudo-triangulations of a point
set.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aarx-clgta-96,
author = {O. Aichholzer and F. Aurenhammer and G. Rote and Y.-F.
Xu},
title = {{{Constant-level greedy triangulations approximate the {MWT}
well}}},
booktitle = {Proc. $2^{nd}$ Int'l. Symp. Operations Research \&
Applications ISORA'96, Lecture Notes in Operations
Research},
pages = {309--318},
year = 1996,
editor = {Du, Zhang, Cheng},
volume = 2,
address = {Guilin, P. R. China},
publisher = {World Publishing Corporation},
category = {3b},
oaich_label = {12b},
postscript = {/files/publications/geometry/aarx-clgta-96.ps.gz},
abstract = {The well-known greedy triangulation $GT(S)$ of a finite
point set $S$ is obtained by inserting compatible edges in
increasing length order, where an edge is compatible if it
does not cross previously inserted ones. Exploiting the
concept of so-called light edges, we introduce a definition
of $GT(S)$ that does not rely on the length ordering of the
edges. Rather, it provides a decomposition of $GT(S)$ into
levels, and the number of levels allows us to bound the
total edge length of $GT(S)$. In particular, we show
$|GT(S)| \leq 3 \cdot 2^{k+1} |MWT(S)|$, where $k$ is the
number of levels and $MWT(S)$ is the minimum-weight
triangulation of $S$.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aarx-ngta-96,
author = {O. Aichholzer and F. Aurenhammer and G. Rote and Y.-F.
Xu},
title = {{{New greedy triangulation algorithms}}},
booktitle = {Proc. $12^{th}$ European Workshop on Computational
Geometry CG '96},
pages = {11--14},
year = 1996,
address = {M{\"u}nster, Germany},
category = {3b},
oaich_label = {11},
postscript = {/files/publications/geometry/aarx-ngta-96.ps.gz},
abstract = {The classical greedy triangulation (GT) of a set $S$ of
$n$ points in the plane is the triangulation obtained by
starting with the empty set (of edges) and at each step
adding the shortest compatible edge between two of the
points of $S$, where a compatible edge is defined to be an
edge that crosses none of the previously added edges. In
this paper we use the greedy method as a general concept to
compute a triangulation of a planar point set. We use
either edges or triangles as basic objects. Furthermore we
give different variants to compute the weight of the
objects, either in a static or dynamic way, leading to a
total of $156$ different greedy triangulation algorithms.
We investigate these algorithms in their quality of
approximating the MWT.},
originalfile = {/geometry/cggg.bib}
}
@article{ar-qdbsb-05,
author = {O. Aichholzer and K. Reinhardt},
title = {{{A quadratic distance bound on sliding between
crossing-free spanning trees}}},
year = 2007,
journal = {Computational Geometry: Theory and Applications, special
issue},
category = {3a},
oaich_label = {53},
pages = {155-161},
volume = {37},
postscript = {/files/publications/geometry/ar-qdbsb-05.ps.gz},
abstract = {Let $S$ be a set of $n$ points in the plane and let
${\mathcal T}_S$ be the set of all crossing-free spanning
trees of $S$. We show that any two trees in ${\mathcal
T}_S$ can be transformed into each other by $O(n^2)$ local
and constant-size edge slide operations. No polynomial
upper bound for this task has been known, but in~\cite{AAH}
a bound of $O(n^2 \log n)$ operations was conjectured.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ahv-cmbps-12,
author = {O.~Aichholzer and F.~Hurtado and B.~Vogtenhuber},
title = {{{Compatible Matchings for Bichromatic Plane Straight-line
Graphs}}},
booktitle = {Proc. $28^{th}$ European Workshop on Computational
Geometry EuroCG '12},
pages = {257--260},
year = 2012,
address = {Assisi, Italy},
category = {3b},
pdf = {/files/publications/geometry/ahv-cmbps-12.pdf},
abstract = {Two plane graphs with the same vertex set are compatible
if their union is again a plane graph. We consider
bichromatic plane straight-line graphs with vertex set $S$
consisting of the same number of red and blue points, and
(perfect) matchings which are compatible to them. For
several different classes $\mathcal{C}$ of graphs, we
present lower and upper bounds such that any given graph
$G(S) \in \mathcal{C}$ admits a compatible (perfect)
matching with this many disjoint edges.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aght-cmgg-11,
author = {O. Aichholzer and A. Garc\'ia and F. Hurtado and J. Tejel},
title = {{{Compatible matchings in geometric graphs}}},
booktitle = {Proc. XIV Encuentros de Geometr\'{\i}a Computacional},
category = {3b},
pages = {145--148},
pdf = {/files/publications/geometry/aght-cmgg-11.pdf},
year = 2011,
address = {Alcal\'a, Spain},
abstract = {Two non-crossing geometric graphs on the same set of
points are compatible if their union is also non-crossing.
In this paper, we prove that every graph G that has an
outerplanar embedding admits a non-crossing perfect
matching compatible with G. Moreover, for non-crossing
geometric trees and simple polygons, we study bounds on the
minimum number of edges that a compatible non-crossing
perfect matching must share with the tree or the polygon.
We also give bounds on the maximal size of a compatible
matching (not necessarily perfect) that is disjoint from
the tree or the polygon.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{haa-nrmwt-97,
author = {R. Hainz and O. Aichholzer and F. Aurenhammer},
title = {{{New results on minimum-weight triangulations and the {LMT}
skeleton}}},
booktitle = {Proc. $13^{th}$ European Workshop on Computational
Geometry CG '97},
pages = {4--6},
year = 1997,
address = {W{\"u}rzburg, Germany},
category = {3b},
oaich_label = {18},
postscript = {/files/publications/geometry/haa-nrmwt-97.ps.gz},
abstract = {Let $P$ be a simple polygon in the plane and let $MWT(P)$
be a minimum-weight triangulation of $P$. We prove that the
$\beta$-skeleton of $P$ is a subset of $MWT(P)$ for all
values $\beta$ > $\sqrt{\frac{4}{3}}$ provided $P$ is
convex or near-convex. This settles the question of
tightness of this bound for a special case and gives
evidence for its validity in the general point set case.\\
We further disprove the conjecture that the so-called
$LMT$-skeleton coincides with the intersection of all
locally minimal triangulations, $LMT(P)$, even for convex
polygons $P$. We introduce an improved $LMT$-skeleton
algorithm which, for simple polygons $P$, exactly computes
$LMT(P)$, and thus a larger subgraph of $MWT(P)$. The
algorithm achieves the same in the general point set case
provided the connectedness of the improved $LMT$-skeleton,
which is given in allmost all practical instances.},
originalfile = {/geometry/cggg.bib}
}
@article{aafrs-mcnmd-13,
author = {B.M.~\'{A}brego and O.~Aichholzer and
S.~Fern\'{a}ndez-Merchant and P.~Ramos and G.~Salazar},
title = {{{More on the crossing number of $K_n$: Monotone drawings}}},
journal = {Electronic Notes in Discrete Mathematics},
volume = {44},
pages = {411--414},
year = 2013,
category = {3b},
note = {Special issue dedicated to LAGOS2013},
doi = {http://dx.doi.org/10.1016/j.endm.2013.10.064},
url = {http://www.sciencedirect.com/science/article/pii/S1571065313002801},
doi = {http://dx.doi.org/10.1016/j.endm.2013.10.064},
abstract = {The Harary-Hill conjecture states that the minimum number
of crossings in a drawing of the complete graph $K_n$ is
$Z(n) :=\frac{1}{4}\left\lfloor \frac{n}{2}\right\rfloor
\left\lfloor \frac{n-1}{2}\right\rfloor \left\lfloor
\frac{n-2}{2}\right\rfloor \left\lfloor
\frac{n-3}{2}\right\rfloor$. This conjecture was recently
proved for 2-page book drawings of $K_n$. As an extension
of this technique, we prove the conjecture for monotone
drawings of $K_n$, that is, drawings where all vertices
have different $x$-coordinates and the edges are $x$-monotone curves.},
originalfile = {/geometry/cggg.bib}
}
@article{aadhru-okcp-11,
author = {O. Aichholzer and F. Aurenhammer and E.D. Demaine and F.
Hurtado and P. Ramos and J. Urrutia},
title = {{{On $k$-convex polygons}}},
journal = {Computational Geometry: Theory and Applications},
year = 2012,
volume = {45(3)},
pages = {73--87},
pdf = {/files/publications/geometry/aadhru-okcp-11.pdf},
abstract = {We introduce the notion of $k$-convexity and explore
polygons in the plane that have this property. Polygons
which are $k$-convex can be triangulated with fast yet
simple algorithms. However, recognizing them is a 3SUM-hard
problem. We give a characterization of 2-convex polygons, a
particularly interesting class, and show how to recognize
them in $O(n \log n)$ time. A description of their shape is
given as well, which leads to Erd{\H{o}}s-Szekeres type
results regarding subconfigurations of their vertex sets.
Finally, we introduce the concept of generalized geometric
permutations, and show that their number can be exponential
in the number of 2-convex objects considered.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aa-chh-94,
author = {O. Aichholzer and F. Aurenhammer},
title = {{{Classifying hyperplanes in hypercubes}}},
booktitle = {Proc. $10^{th}$ European Workshop on Computational
Geometry CG '94},
pages = {53--57},
year = 1994,
address = {Santander, Spain},
category = {3b},
oaich_label = {3a},
postscript = {/files/publications/geometry/aa-chh-94.ps.gz},
abstract = {We consider hyperplanes spanned by vertices of the unit
$d$-cube. We classify these hyperplanes by parallelism to
coordinate axes, by symmetry of the $d$-cube vertices they
avoid, as well as by so-called hull-honesty. (Hull-honest
hyperplanes are those whose intersection figure with the
$d$-cube coincides with the convex hull of the $d$-cube
vertices they contain; they do not cut $d$-cube edges
properly.) We describe relationships between these classes,
and give the exact number of hull-honest hyperplanes, in
general dimensions. An experimental enumeration of all
spanned hyperplanes up to dimension eight showed us the
intrinsic difficulty of developing a general enumeration
scheme. Motivation for considering such hyperplanes stems
from coding theory, from linear programming, and from the
theory of machine learning.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ak-aoten-05b,
author = {O. Aichholzer and H. Krasser},
title = {{{Abstract Order Type Extension and New Results on the
Rectilinear Crossing Number}}},
booktitle = {Proc. $21^{th}$ Ann. ACM Symp. Computational Geometry},
year = 2005,
pages = {91--98},
address = {Pisa, Italy},
category = {3b},
oaich_label = {54b},
postscript = {/files/publications/geometry/ak-aoten-05b.ps.gz},
abstract = {We extend the order type data base of all realizable order
types in the plane to point sets of cardinality 11. More
precisely, we provide a complete data base of all
combinatorial different sets of up to 11 points in general
position in the plane. Moreover we develop a novel and
efficient method for a complete extension to order types of
size 12 and more in an abstract sense, that is, without the
need to store or realize the sets. The presented method is
well suited for independent computations and thus time
intensive investigations benefit from the possibility of
distributed computing.\\ Our approach has various
applications to combinatorial problems which are based on
sets of points in the plane. This includes classic problems
like searching for (empty) convex k-gons ('happy end problem'),
decomposing sets into convex regions, counting
structures like triangulations or pseudo-triangulations,
minimal crossing numbers, and more. We present some
improved results to all these problems. As an outstanding
result we have been able to determine the exact rectilinear
crossing number for up to $n=17$, the largest previous
range being $n=12$, and slightly improved the asymptotic
upper bound. },
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aaks-cmpt-02,
author = {O. Aichholzer and F. Aurenhammer and H. Krasser and B.
Speckmann},
title = {{{Convexity Minimizes Pseudo-Triangulations}}},
booktitle = {Proc. $14th$ Annual Canadian Conference on Computational
Geometry CCCG 2002},
pages = {158--161},
year = 2002,
address = {Lethbridge, Alberta, Canada},
category = {3b},
oaich_label = {42},
postscript = {/files/publications/geometry/aaks-cmpt-02.ps.gz},
abstract = {For standard triangulations it is not known which sets of
points have the fewest or the most triangulations. In
contrast, we show that sets of points in convex position
minimize the number of minimum pseudo-triangulations. This
adds to the common belief that minimum
pseudo-triangulations are more tractable in many
respects.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aaabbcilsty-sseps-13,
author = {O. Aichholzer and S.R. Allen and G. Aloupis and L. Barba
and P. Bose and J.-L. De Carufel and J. Iacono and S.
Langerman and D.L. Souvaine and P. Taslakin and M.
Yagnatinsky},
title = {{{Sum of Squared Edges for MST of a Point Set in a Unit
Square}}},
booktitle = {Proc. of the $16^{th}$ Japan Conference on Discrete and
Computational Geometry and Graphs (JCDCG$^2$ 2013)},
year = 2013,
address = {Tokyo, Japan},
category = {3b},
abstract = {Given a set P of points in the unit square let w(P) be the
minimum sum of the squares of the edge lengths in the
minimum spanning tree of P. We show that w(P) < 3.411.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aahru-tcp-09,
author = {O. Aichholzer and F. Aurenhammer and F.~Hurtado and
P.~Ramos and J.~Urrutia},
title = {{{Two-Convex Polygons}}},
booktitle = {Proc. $25^{th}$ European Workshop on Computational
Geometry EuroCG '09},
category = {3b},
pages = {117--120},
pdf = {/files/publications/geometry/aahru-tcp-09.pdf},
oaich_label = {82},
year = 2009,
address = {Brussels, Belgium},
abstract = {We introduce a notion of $k$-convexity and explore some
properties of polygons that have this property. In
particular, \mbox{$2$-convex} polygons can be recognized in
\mbox{$O(n \log n)$} time, and \mbox{$k$-convex} polygons
can be triangulated in $O(kn)$ time.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aahv-gceps-06,
author = {O. Aichholzer and F. Aurenhammer and C. Huemer and B.
Vogtenhuber},
title = {{{Gray code enumeration of plane straight-line graphs}}},
booktitle = {Proc. $22^{nd}$ European Workshop on Computational
Geometry EuroCG '06},
pages = {71--74},
category = {3b},
oaich_label = {62},
year = 2006,
address = {Delphi, Greece},
postscript = {/files/publications/geometry/aahv-gceps-06.ps.gz},
htmlnote = {Also available as FSP-report S092-7, Austria, 2006, at http://www.industrial-geometry.at/techrep.php.},
abstract = {We develop Gray code enumeration schemes for geometric
straight-line graphs in the plane. The considered graph
classes include plane graphs, connected plane graphs, and
plane spanning trees. Previous results were restricted to
the case where the underlying vertex set is in convex
position.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ahn-ntepp-01,
author = {O. Aichholzer and F. Hurtado and M. Noy},
title = {{{On the Number of Triangulations Every Planar Point Set
Must Have}}},
booktitle = {Proc. $13th$ Annual Canadian Conference on Computational
Geometry CCCG 2001},
pages = {13--16},
year = 2001,
address = {Waterloo, Ontario, Canada},
category = {3b},
oaich_label = {33},
postscript = {/files/publications/geometry/ahn-ntepp-01.ps.gz},
htmlnote = {See also the Counting
Triangulations - Olympics.},
abstract = {We show that the number of straight line triangulations
exhibited by any set of $n$ points in general position in
the plane is bounded from below by
$\Omega((2+\varepsilon)^n)$ for some $\varepsilon > 0$. To
the knowledge of the authors this is the first non-trivial
lower bound.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aaddfhlsw-cfs-13,
author = {O.~Aichholzer and G.~Aloupis and E.D.~Demaine and
M.L.~Demaine and S.P.~Fekete and M.~Hoffmann and A.~Lubiwk
and J.~Snoeyink and a.~Winslow},
title = {{{Covering Folded Shapes}}},
booktitle = {Proc. $25^{th}$ Annual Canadian Conference on
Computational Geometry CCCG 2013},
pages = {73--78},
year = 2013,
address = {Waterloo, Ontario, Canada},
category = {3b},
htmlnote = {proceedings},
abstract = {Can folding a piece of paper flat make it larger? In this
paper, we explore how large a shape $S$ must be scaled to
cover a flat-folded copy of itself. We consider both single
folds and arbitrary folded states. The underlying problem
is motivated by computational origami, but also related to
other types of covering problems. In addition to
considering special shapes (squares, equilateral triangles,
polygons and disks), we give necessary and sufficient
scaling factors for single folds to convex objects and
arbitrary folds to simply-connected objects.},
originalfile = {/geometry/cggg.bib}
}
@article{aahk-tct-01b,
author = {O. Aichholzer and F. Aurenhammer and F. Hurtado and H.
Krasser},
title = {{{Towards Compatible Triangulations}}},
year = 2003,
journal = {Theoretical Computer Science},
note = {Special Issue},
volume = {296},
pages = {3--13},
publisher = {Elsevier},
category = {3a},
oaich_label = {32b},
postscript = {/files/publications/geometry/aahk-tct-01b.ps.gz},
abstract = {We state the following conjecture: any two planar
$n$-point sets (that agree on the number of convex hull
points) can be triangulated in a compatible manner, i.e.,
such that the resulting two triangulations are
topologically equivalent. The conjecture is proved true for
point sets with at most three interior points. We further
exhibit a class of point sets which can be triangulated
compatibly with any other set that satisfies the obvious
size and hull restrictions. Finally, we prove that adding a
small number of extraneous points (the number of interior
points minus two) always allows for compatible
triangulations.},
originalfile = {/geometry/cggg.bib}
}
@article{agor-nrlbn-07,
author = {O. Aichholzer and J. Garc\'{\i}a and D. Orden and P.A.
Ramos},
title = {New results on lower bounds for the number of~$(\leq
k)$-facets},
journal = {European Journal of Combinatorics},
year = 2009,
volume = {30},
category = {3a},
number = {},
oaich_label = {67},
pages = {1568--1574},
postscript = {/files/publications/geometry/agor-nrlbn-07.pdf},
abstract = {In this paper we present three different results dealing
with the number of $(\leq k)$-facets of a set of points:
(1) We give structural properties of sets in the plane that
achieve the optimal lower bound $3{k+2 \choose 2}$ of
$(\leq k)$-edges for a fixed $k\leq \lfloor n/3 \rfloor
-1$; (2) We show that the lower bound $3{k+2 \choose
2}+3{k-\lfloor \frac{n}{3} \rfloor+2 \choose 2}$ for the
number of $(\leq k)$-edges of a planar point set is optimal
in the range $\lfloor n/3 \rfloor \leq k \leq \lfloor 5n/12
\rfloor -1$; (3) We show that, for $k < n/4$, the number of
$(\leq k)$-facets a set of $n$ points in $R^3$ in general
position is at least $4{k+3 \choose 3}$, and that this
bound is tight in that range.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aor-ssarc-06,
author = {O. Aichholzer and D. Orden and P.A. Ramos},
title = {{{On the Structure of sets attaining the rectilinear
crossing number}}},
booktitle = {Proc. $22^{nd}$ European Workshop on Computational
Geometry EuroCG '06},
pages = {43--46},
category = {3b},
oaich_label = {63},
year = 2006,
address = {Delphi, Greece},
postscript = {/files/publications/geometry/aor-ssarc-06.pdf},
abstract = {We study the structural properties of the point
configurations attaining the rectilinear crossing number
$\overline{cr}(K_n)$, that is, those $n$-point sets that
minimize the number of crossings over all possible
straight-edge embeddings of $K_n$ in the plane. As a main
result we prove the conjecture that such sets always have a
triangular convex hull. The techniques developed allow us
to show a similar result for the halving-edge problem: For
any $n$ there exists a set of $n$ points with triangular
convex hull that maximizes the number of halving edges.
Moreover, we provide a simpler proof of the following
result: any set of points in the plane in general position
has at least $3{j+2 \choose 2}$ $(\leq j)$-edges. This
bound is known to be tight for $0\leq j\leq
\lfloor\frac{n}{3}\rfloor-1$. In addition, we show that for
point sets achieving this bound the
$\lfloor\frac{n+3}{6}\rfloor$ outermost convex layers are triangles.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aabk-sept-03,
author = {O. Aichholzer and F. Aurenhammer and P. Brass and H.
Krasser},
title = {{{Spatial Embedding of Pseudo-Triangulations}}},
booktitle = {Proc. $19^{th}$ Ann. ACM Symp. Computational Geometry},
address = {San Diego, California, USA},
volume = 19,
pages = {144--153},
category = {3b},
year = 2003,
oaich_label = {45},
abstract = {We show that pseudo-triangulations have natural embeddings
in three-space. As a consequence, various concepts for
triangulations, like flipping to optimality, (constrained)
Delaunayhood, and a polytope representation carry over to
pseudo-triangulations.},
postscript = {/files/publications/geometry/aabk-sept-03.ps.gz},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aaclmp-vddsd-97,
author = {O. Aichholzer and F. Aurenhammer and D.Z. Chen and D.T.
Lee and A. Mukhopadhyay and E. Papadopoulou},
title = {{{{V}oronoi diagrams for direction-sensitive distances
(communication)}}},
booktitle = {Proc. $13^{th}$ Ann. ACM Symp. Computational Geometry},
pages = {418--420},
year = 1997,
address = {Nice, France},
category = {3b},
note = {[SFB Report F003-098, TU Graz, Austria, 1996]},
oaich_label = {16},
postscript = {/files/publications/geometry/aaclmp-vddsd-97.ps.gz},
abstract = {On a tilted plane $T$ in three-space, direction-sensitive
distances are defined as the Euclidean distance plus a
multiple of the signed difference in height. These
direction-sensitive distances, called skew distances
generalize the Euclidean distance and may model realistic
environments more closely than the Euclidean distance.
Various Voronoi diagrams and related problems under this
kind of distances are investigated. A relationship to
convex distance functions and to Euclidean Voronoi diagrams
for planar circles is shown, and is exploited for a
geometric analysis and a plane-sweep construction of
Voronoi diagrams on $T$. Several optimal algorithms based
on the direction-sensitive distances on $T$ are presented.
For example, an output-sensitive algorithm is developed for
computing the skew distance Voronoi diagram with $n$ sites
on $T$, in $O(n \log h)$ time and $O(n)$ space, where $h$
is the number of sites which have non-empty Voronoi regions
($1 \leq h \leq n$). $O(n \log n)$ time and $O(n)$ space
algorithms are also given for several other problems under
skew distances, including the all nearest neighbors and the
layers of Voronoi diagram. These algorithms have certain
features different from their 'ordinary' counterparts based
on the Euclidean distance.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aaadj-at-10,
author = {O.~Aichholzer and W.~Aigner and F.~Aurenhammer and
K.\v{C}~Dobi\'a\v{s}ov\'a and B.~J\"uttler},
title = {{{Arc Triangulations}}},
booktitle = {Proc. $26^{th}$ European Workshop on Computational
Geometry EuroCG '10},
pages = {17--20},
year = 2010,
address = {Dortmund, Germany},
pdf = {/files/publications/geometry/aaadj-at-10.pdf},
abstract = {The quality of a triangulation is, in many practical
applications, influenced by the angles of its triangles. In
the straight line case, angle optimization is not possible
beyond the Delaunay triangulation. We propose and study the
concept of circular arc triangulations, a simple and
effective alternative that offers flexibility for
additionally enlarging small angles. We show that angle
optimization and related questions lead to linear
programming problems, and we define unique flips in arc
triangulations. Moreover, applications of certain classes
of arc triangulations in the areas of finite element
methods and graph drawing are sketched.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{a-ecgrr-09,
author = {O. Aichholzer},
title = {{{[{E}mpty] [colored] $k$-gons - {R}ecent results on some
{E}rd\"os-{S}zekeres type problems}}},
booktitle = {Proc. XIII Encuentros de Geometr\'{\i}a Computacional},
category = {3b},
pages = {43--52},
pdf = {/files/publications/geometry/a-ecgrr-09.pdf},
oaich_label = {85},
year = 2009,
address = {Zaragoza, Spain},
abstract = {We consider a family of problems which are based on a
question posed by Erd{\H{o}}s and Szekeres in 1935: ``What is
the smallest integer $g(k)$ such that any set of $g(k)$
points in the plane contains at least one convex $k$-gon?''
In the mathematical history this has become well known as
the ``Happy End Problem''. There are several variations of
this problem: The $k$-gons might be required to be empty,
that is, to not contain any points of the set in their
interior. In addition the points can be colored, and we
look for monochromatic $k$-gons, meaning polygons spanned
by points of the same color. Beside the pure existence
question we are also interested in the asymptotic behavior,
for example whether there are super-linear many $k$-gons of
some type. And finally, for several of these problems even
small non-convex $k$-gons are of interest. We will survey
recent progress and discuss open questions for this class
of problems.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ahk-twpst-04,
author = {O. Aichholzer and C. Huemer and H. Krasser},
title = {{{Triangulations Without Pointed Spanning Trees - Extended
Abstract}}},
booktitle = {Proc. $20^{th}$ European Workshop on Computational
Geometry EWCG '04},
year = 2004,
pages = {221--224},
address = {Sevilla, Spain},
category = {3b},
oaich_label = {50},
postscript = {/files/publications/geometry/ahk-twpst-04.ps.gz},
abstract = {Problem $50$ in the Open Problems Project~\cite{OPP} asks
whether any triangulation on a point set in the plane
contains a pointed spanning tree as a subgraph. We provide
a counterexample. As a consequence we show that there exist
triangulations which require a linear number of edge flips
to become Hamiltonian. },
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ak-aoten-05a,
author = {O. Aichholzer and H. Krasser},
title = {{{Abstract Order Type Extension and New Results on the
Rectilinear Crossing Number}}},
booktitle = {Proc. $21^{th}$ European Workshop on Computational
Geometry EWCG '05},
year = 2005,
pages = {61--64},
address = {Eindhoven, The Nederlands},
category = {3b},
oaich_label = {54},
postscript = {/files/publications/geometry/ak-aoten-05a.ps.gz},
pdf = {/files/publications/geometry/ak-aoten-05a.pdf},
abstract = {We extend the order type data base of all realizable order
types in the plane to point sets of cardinality 11. More
precisely, we provide a complete data base of all
combinatorial different sets of up to 11 points in general
position in the plane. Moreover we develop a novel and
efficient method for a complete extension to order types of
size 12 and more in an abstract sense, that is, without the
need to store or realize the sets. The presented method is
well suited for independent computations and thus time
intensive investigations benefit from the possibility of
distributed computing.\\ Our approach has various
applications to combinatorial problems which are based on
sets of points in the plane. This includes classic problems
like searching for (empty) convex k-gons('happy end problem'),
decomposing sets into convex regions, counting
structures like triangulations or pseudo-triangulations,
minimal crossing numbers, and more. We present some
improved results to all these problems. As an outstanding
result we have been able to determine the exact rectilinear
crossing number for up to $n=17$, the largest previous
range being $n=12$, and slightly improved the asymptotic
upper bound. },
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aafls-tucmr-11,
author = {B.M.~{\'A}brego and O.~Aichholzer and
S.~Fern{\'a}ndez-Merchant and J.~Lea{\~n}os and G.~Salazar},
title = {{{There is a unique crossing-minimal rectilinear drawing of
$K_{18}$}}},
booktitle = {Electronic Notes in Discrete Mathematics},
category = {3b},
pages = {547-552},
volume = {38},
pdf = {/files/publications/geometry/aafls-tucmr-11.pdf},
year = 2011,
abstract = {We show that, up to isomorphism, there is a unique
crossing-minimal rectilinear drawing of $K_{18}$. As a
consequence we settle, in the negative, the following
question from Aichholzer and Krasser: does there always
exist an crossing-minimal drawing of $K_n$ that contains a
crossing-minimal drawing of $K_{n-1}$?},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aa-ssgpf-96,
author = {O. Aichholzer and F. Aurenhammer},
title = {{{Straight skeletons for general polygonal figures}}},
booktitle = {Proc. $2^{nd}$ Ann. Int'l. Computing and Combinatorics
Conf. COCOON'96, Lecture Notes in Computer Science},
pages = {117--126},
year = 1996,
volume = 1090,
address = {Hong Kong},
publisher = {Springer Verlag},
category = {3b},
oaich_label = {10},
note = {[IIG-Report-Series 423, TU Graz, Austria, 1995]},
postscript = {/files/publications/geometry/aa-ssgpf-96.ps.gz},
abstract = {A novel type of skeleton for general polygonal figures,
the straight skeleton $S(G)$ of a planar straight line
graph $G$, is introduced and discussed. Exact bounds on the
size of $S(G)$ are derived. The straight line structure of
$S(G)$ and its lower combinatorial complexity may make
$S(G)$ preferable to the widely used Voronoi diagram (or
medial axis) of $G$ in several applications. We explain why
$S(G)$ has no Voronoi diagram based interpretation and why
standard construction techniques fail to work. A simple
$O(n)$ space algorithm for constructing $S(G)$ is proposed.
The worst-case running time is $O(n^3 \log n)$, but the
algorithm can be expected to be practically efficient, and
it is easy to implement. We also show that the concept of
$S(G)$ is flexible enough to allow an individual weighting
of the edges and vertices of $G$, without changes in the
maximal size of $S(G)$, or in the method of construction.
Apart from offering an alternative to Voronoi-type
skeletons, these generalizations of $S(G)$ have
applications to the reconstruction of a geographical
terrain from a given river map, and to the construction of
a polygonal roof above a given layout of ground walls.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aak-cncg-02,
author = {O. Aichholzer and F. Aurenhammer and H. Krasser},
title = {{{On the Crossing Number of Complete Graphs}}},
year = 2002,
booktitle = {Proc. $18^{th}$ Ann. ACM Symp. Computational Geometry},
pages = {19--24},
address = {Barcelona, Spain},
category = {3b},
oaich_label = {36b},
htmlnote = {See also our crossing
number homepage.},
postscript = {/files/publications/geometry/aak-cncg-02.ps.gz},
abstract = {Let $\overline{cr}(G)$ denote the rectilinear crossing
number of a graph $G$. We determine
$\overline{cr}(K_{11})=102$ and
$\overline{cr}(K_{12})=153$. Despite the remarkable hunt
for crossing numbers of the complete graph~$K_n$ --
initiated by R.~Guy in the 1960s -- these quantities have
been unknown for $n>10$ to~date. Our solution mainly relies
on a tailor-made method for enumerating all inequivalent
sets of points (so-called order types) of size $11$. Based
on these findings, we establish new upper and lower bounds
on $\overline{cr}(K_{n})$ for general~$n$. Specific values
for $n \leq 45$ are given, along with significantly
improved asymptotic values. The asymptotic lower bound is
immediate from the fact $\overline{cr}(K_{11})=102$,
whereas the upper bound stems from a novel construction of
drawings with few crossings. The construction is shown to
be optimal within its frame. The tantalizing question of
determining $\overline{cr}(K_{13})$ is left open. The
latest ra(n)ge is $\{221,223,225,227,229\}$; our conjecture
is $\overline{cr}(K_{13}) = 229$.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aaabbli-ssemp-12,
author = {O. Aichholzer and S.R. Allen and G. Aloupis and L. Barba
and P. Bose and S. Langerman and J. Iacono},
title = {{{Sum of Squared Edges for MST of a Point Set in a Unit
Square}}},
booktitle = {Proc. $22^{nd}$ Annual Fall Workshop on Computational
Geometry},
year = 2012,
address = {University of Maryland, Maryland, USA},
category = {3b},
htmlnote = {For
proceedings see here.},
abstract = {Given a set P of points in the unit square let w(P) be the
minimum sum of the squares of the edge lengths in the
minimum spanning tree of P. We show that w(P) < 3.411.},
originalfile = {/geometry/cggg.bib}
}
@article{aa-vdcgf-02,
author = {O. Aichholzer and F. Aurenhammer},
title = {{{Voronoi Diagrams - Computational Geometry's Favorite}}},
journal = {Special Issue on Foundations of Information Processing of
{TELEMATIK}},
pages = {7--11},
volume = 1,
year = {2002},
address = {Graz, Austria},
category = {4a},
oaich_label = {39},
postscript = {/files/publications/geometry/aa-vdcgf-02.ps.gz},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{agor-nlbnk-06b,
author = {O. Aichholzer and J. Garc\'{\i}a and D. Orden and P.A.
Ramos},
title = {New lower bounds for the number of~$(\leq k)$-edges and
the rectilinear crossing number of~$K_n$},
year = 2006,
category = {3a},
oaich_label = {64b},
pages = {57--64},
booktitle = {Actas de las IV Jornadas de Matematica Discreta y
Algoritmica},
postscript = {/files/publications/geometry/agor-nlbnk-06.ps.gz},
abstract = {We provide a new lower bound on the number of~$(\leq
k)$-edges on a set of~$n$ points in the plane in general
position. We show that for $0 \leq k \leq
\lfloor\frac{n-2}{2}\rfloor$ the number of~$(\leq k)$-edges
is at least $ E_k(S) \geq 3 {k+2 \choose 2} +
\sum_{j=\lfloor\frac{n}{3}\rfloor}^k (3j-n+3)$, which, for
$k\geq \lfloor \frac{n}{3}\rfloor$, improves the previous
best lower bound.\\ As a main consequence, we obtain a new
lower bound on the rectilinear crossing number of the
complete graph or, in other words, on the minimum number of
convex quadrilaterals determined by~$n$ points in the plane
in general position. We show that the crossing number is at
least $ \left(\frac{41}{108}+\varepsilon \right) {n \choose
4} + O(n^3) \geq 0.379631 {n \choose 4} + O(n^3)$, which
improves the previous bound of~$0.37533 {n \choose 4} +
O(n^3)$ and approaches the best known upper bound $0.38058
{n \choose 4}$.\\ The proof is based on a result about the
structure of sets attaining the rectilinear crossing
number, for which we show that the convex hull is always a
triangle.\\ Further implications include improved results
for small values of $n$. We extend the range of known
values for the rectilinear crossing number, namely by
$cr(K_{19})=1318$ and $cr(K_{21})=2055$. Moreover we
provide improved upper bounds on the maximum number $h_n$
of halving edges a point set can have.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{acddemoprt-fp-00a,
author = {O. Aichholzer and C. Cort\'{e}s and E.D. Demaine and V.
Dujmovi\'{c} and J. Erickson and H. Meijer and M. Overmars
and B. Palop and S. Ramaswami and G.T. Toussaint},
title = {{{Flipturning Polygons}}},
booktitle = {Proc. Japan Conference on Discrete and Computational
Geometry JCDCG 2000},
year = 2000,
address = {Tokay University, Tokyo, Japan},
category = {3b},
oaich_label = {27},
postscript = {/files/publications/geometry/acddemoprt-fp-00a.ps.gz},
abstract = {A flipturn is an operation that transforms a nonconvex
simple polygon into another simple polygon, by rotating a
concavity 180 degrees around the midpoint of its bounding
convex hull edge. Joss and Shannon proved in 1973 that a
sequence of flipturns eventually transforms any simple
polygon into a convex polygon. This paper describes several
new results about such flipturn sequences. We show that any
orthogonal polygon is convexified after at most $n-5$
arbitrary flipturns, or at most $5(n-4)/6$ well-chosen
flipturns, improving the previously best upper bound of
$(n-1)!/2$. We also show that any simple polygon can be
convexified by at most $n^2-4n+1$ flipturns, generalizing
earlier results of Ahn et al. These bounds depend
critically on how degenerate cases are handled; we
carefully explore several possibilities. We describe how to
maintain both a simple polygon and its convex hull in
$O(\log^4 n)$ time per flipturn, using a data structure of
size $O(n)$. We show that although flipturn sequences for
the same polygon can have very different lengths, the shape
and position of the final convex polygon is the same for
all sequences and can be computed in $O(n \log n)$ time.
Finally, we demonstrate that finding the longest
convexifying flipturn sequence of a simple polygon is
NP-hard.},
originalfile = {/geometry/cggg.bib}
}
@article{aaag-ntsp-95,
author = {O. Aichholzer and D. Alberts and F. Aurenhammer and B.
G{\"a}rtner},
title = {{{A novel type of skeleton for polygons}}},
journal = {Journal of Universal Computer Science},
year = 1995,
volume = 1,
number = 12,
pages = {752--761},
htmlnote = {Click here for the Online
Version},
category = {3a},
oaich_label = {9},
note = {[IIG-Report-Series 424, TU Graz, Austria, 1995]},
postscript = {/files/publications/geometry/aaag-ntsp-95.ps.gz},
abstract = {A new internal structure for simple polygons, the straight
skeleton, is introduced and discussed. It is composed of
pieces of angular bisectores which partition the interior
of a given $n$-gon $P$ in a tree-like fashion into $n$
monotone polygons. Its straight-line structure and its
lower combinatorial complexity may make the straight
skeleton preferable to the widely used medial axis of a
polygon. As a seemingly unrelated application, the straight
skeleton provides a canonical way of constructing a
polygonal roof above a general layout of ground walls.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aak-cncg-02b,
author = {O. Aichholzer and F. Aurenhammer and H. Krasser},
title = {{{On the Crossing Number of Complete Graphs - Extended
Abstract}}},
year = 2002,
booktitle = {Proc. $18^{th}$ European Workshop on Computational
Geometry CG '02 Warszawa},
pages = {90-92},
address = {Warszawa, Poland},
category = {3b},
oaich_label = {36c},
htmlnote = {See also our crossing
number homepage.},
postscript = {/files/publications/geometry/aak-cncg-02b.ps.gz},
abstract = {Let $\overline{cr}(G)$ denote the rectilinear crossing
number of a graph $G$. We determine
$\overline{cr}(K_{11})=102$ and
$\overline{cr}(K_{12})=153$. Despite the remarkable hunt
for crossing numbers of the complete graph~$K_n$ --
initiated by R.~Guy in the 1960s -- these quantities have
been unknown for $n>10$ to~date. Our solution mainly relies
on a tailor-made method for enumerating all inequivalent
sets of points (so-called order types) of size $11$. Based
on these findings, we establish new upper and lower bounds
on $\overline{cr}(K_{n})$ for general~$n$. Specific values
for $n \leq 45$ are given, along with significantly
improved asymptotic values. The asymptotic lower bound is
immediate from the fact $\overline{cr}(K_{11})=102$,
whereas the upper bound stems from a novel construction of
drawings with few crossings. The construction is shown to
be optimal within its frame. The tantalizing question of
determining $\overline{cr}(K_{13})$ is left open. The
latest ra(n)ge is $\{221,223,225,227,229\}$; our conjecture
is $\overline{cr}(K_{13}) = 229$.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{abdgh-cgm-08,
author = {O. Aichholzer and S. Bereg and A. Dumitrescu and A.
Garc\'{\i}a and C. Huemer and F. Hurtado and M. Kano and A.
M{\'a}rquez and D. Rappaport and S. Smorodinsky and D.
Souvaine and J. Urrutia and D. Wood},
title = {{{Compatible Geometric Matchings}}},
booktitle = {Proc. $1st$ Topological \& Geometric Graph Theory 2008},
pages = {194--199},
year = 2008,
address = {Paris, France},
category = {3b},
oaich_label = {76},
postscript = {/files/publications/geometry/abdgh-cgm-08.pdf},
abstract = {Abstract: This paper studies non-crossing geometric
perfect matchings. Two such perfect matchings are
compatible if they have the same vertex set and their union
is also non-crossing. Our first result states that for any
two perfect matchings $M$ and $M'$ of the same set of $n$
points, for some $k \in O(log n)$, there is a sequence of
perfect matchings $M = M_0,M_1, . . . ,M_k = M'$, such that
each $M_i$ is compatible with $M_{i+1}$. This improves the
previous best bound of $k \leq n-2$. We then study the
conjecture: every perfect matching with an even number of
edges has an edge-disjoint compatible perfect matching. We
introduce a sequence of stronger conjectures that imply
this conjecture, and prove the strongest of these
conjectures in the case of perfec matchings that consist of
vertical and horizontal segments. Finally, we prove that
every perfect matching with $n$ edges has an edge-disjoint
compatible matching with approximately $4n/5$ edges.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{agor-nrlbn-07b,
author = {O. Aichholzer and J. Garc\'{\i}a and D. Orden and P.A.
Ramos},
title = {New results on lower bounds for the number of~$(\leq
k)$-facets},
booktitle = {Proceedings EuroComb'07, Electronic Notes in Discrete
Mathematics},
pages = {189--193},
year = 2007,
volume = {29C},
category = {3a},
oaich_label = {67b},
postscript = {/files/publications/geometry/agor-nrlbn-07.pdf},
abstract = {In this paper we present three different results dealing
with the number of $(\leq k)$-facets of a set of points:
(1) We give structural properties of sets in the plane that
achieve the optimal lower bound $3{k+2 \choose 2}$ of
$(\leq k)$-edges for a fixed $k\leq \lfloor n/3 \rfloor
-1$; (2) We show that the lower bound $3{k+2 \choose
2}+3{k-\lfloor \frac{n}{3} \rfloor+2 \choose 2}$ for the
number of $(\leq k)$-edges of a planar point set is optimal
in the range $\lfloor n/3 \rfloor \leq k \leq \lfloor 5n/12
\rfloor -1$; (3) We show that, for $k < n/4$, the number of
$(\leq k)$-facets a set of $n$ points in $R^3$ in general
position is at least $4{k+3 \choose 3}$, and that this
bound is tight in that range.},
originalfile = {/geometry/cggg.bib}
}
@article{ahn-lbntp-04,
author = {O. Aichholzer and F. Hurtado and M. Noy},
title = {{{A Lower Bound on the Number of Triangulations of Planar
Point Sets}}},
year = 2004,
journal = {Computational Geometry: Theory and Applications},
volume = {29},
number = {2},
pages = {135--145},
category = {3a},
oaich_label = {52},
postscript = {/files/publications/geometry/ahn-lbntp-04.ps.gz},
pdf = {/files/publications/geometry/ahn-lbntp-04.pdf},
htmlnote = {See also the Counting
Triangulations - Olympics.},
abstract = {We show that the number of straight-edge triangulations
exhibited by any set of $n$ points in general position in
the plane is bounded from below by $\Omega(2.33^n)$.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aah-eoncs-00,
author = {O. Aichholzer and F. Aurenhammer and F. Hurtado},
title = {{{Edge Operations on Non-Crossing Spanning Trees}}},
booktitle = {Proc. $16^{th}$ European Workshop on Computational
Geometry CG '2000},
pages = {121--125},
year = 2000,
address = {Eilat, Israel},
category = {3b},
oaich_label = {23},
postscript = {/files/publications/geometry/aah-eoncs-00.ps.gz},
htmlnote = {You can download our MST-Tool.},
abstract = {Let $S$ be a set of $n$ points in the Euclidean plane.
Consider the set ${\cal T}_S$ of all non-crossing spanning
trees of $S$. A {\em tree graph\/} ${\cal TG}_{\tt op}(S)$
is the graph that has ${\cal T}_S$ as its vertex set and
that connects vertex (tree) $T$ to vertex $T'$ iff $T' =
{\tt op}(T)$, where ${\tt op}$ is some operation that
exchanges two tree edges following a specific rule. The
existence of a path between two vertices in ${\cal TG}_{\tt
op}(S)$ means transformability of the corresponding trees
into each other by repeated application of the operation
${\tt op}$. The length of a shortest path corresponds to
the distance between the two trees with respect to the
operation ${\tt op}$. Distances of this kind provide a
measure of similarity between trees. We prove new results
on ${\cal TG}_{\tt op}(S)$ for two classical operations
${\tt op}$, namely the (improving and crossing-free) {\em
edge move\/} and the (crossing-free) {\em edge slide\/}.
Applications to morphing of trees and to the continuous
deformation of sets of line segments seem reasonable. Our
results mainly rely on a fact of interest in its own right:
Let $MST(S)$ and $DT(S)$ be the minimum spanning tree and
the Delaunay triangulation of $S$, respectively. Then any
pair $(T,\Delta)$, for $T \in {\cal T}_S$ and $\Delta$
being $T's$ constrained Delaunay triangulation, can be
transformed into the pair $(MST(S),DT(S))$ via a canonical
tree/triangulation sequence.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{abbbkrtv-t3c-13,
author = {Oswin Aichholzer and Sang Won Bae and Luis Barba and
Prosenjit Bose and Matias Korman and Andr{\'e} van Renssen
and Perouz Taslakian and Sander Verdonschot},
title = {{{Theta 3 is connected}}},
booktitle = {Proc. $25^{th}$ Annual Canadian Conference on
Computational Geometry CCCG 2013},
pages = {205--211},
year = 2013,
address = {Waterloo, Ontario, Canada},
category = {3b},
note = {invited to special issue of CCCG 2013},
htmlnote = {proceedings},
abstract = {In this paper, we show that the $\theta$-graph with three
cones is connected. We also provide an alternative proof of
the connectivity of the Yao-graph with three cones.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aak-eotsp-01,
author = {O. Aichholzer and F. Aurenhammer and H. Krasser},
title = {{{Enumerating Order Types for Small Point Sets with
Applications}}},
booktitle = {Proc. $17^{th}$ Ann. ACM Symp. Computational Geometry},
pages = {11--18},
year = 2001,
address = {Medford, Massachusetts, USA},
category = {3b},
oaich_label = {31},
postscript = {/files/publications/geometry/aak-eotsp-01.ps.gz},
htmlnote = {See also our order
type homepage.},
abstract = {Order types are a means to characterize the combinatorial
properties of a finite point configuration. In particular,
the crossing properties of all straight-line segments
spanned by a planar $n$-point set are reflected by its
order type. We establish a complete and reliable data base
for all possible order types of size $n=10$ or less. The
data base includes a realizing point set for each order
type in small integer grid representation. To our
knowledge, no such project has been carried out before. We
substantiate the usefulness of our data base by applying it
to several problems in computational and combinatorial
geometry. Problems concerning triangulations, simple
polygonalizations, complete geometric graphs, and $k$-sets
are addressed. This list of possible applications is not
meant to be exhaustive. We believe our data base to be of
value to many researchers who wish to examine their
conjectures on small point configurations. },
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aaag-sssp-95,
author = {O. Aichholzer and D. Alberts and F. Aurenhammer and B.
G{\"a}rtner},
title = {{{Straight skeletons of simple polygons}}},
booktitle = {Proc. $4^{th}$ Int. Symp. of LIESMARS},
pages = {114--124},
year = 1995,
address = {Wuhan, P. R. China},
category = {3b},
oaich_label = {9a},
postscript = {/files/publications/geometry/aaag-sssp-95.ps.gz},
abstract = {A new internal structure for simple polygons, the straight
skeleton, is introduced and discussed. It is a tree and
partitions the interior of a given $n$-gon $P$ into $n$
monotone polygons, one for each edge of $P$. Its
straight-line structure and its lower combinatorial
complexity may make the straight skeleton $S(P)$ preferable
to the widely used medial axis of $P$. We show that $S(P)$
has no Voronoi diagram structure and give an $O(n r \log
n)$ time and $O(n)$ space construction algorithm, where $r$
counts the reflex vertices of $P$. As a seemingly unrelated
application, the straight skeleton provides a canonical way
of constructing a roof of given slope above a polygonal
layout of ground walls.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{acdfon-eapp-13,
author = {O.~Aichholzer and L.E.~Caraballo and
J.M.~D\'{\i}az-B{\'a}{\~n}ez and R.~Fabila-Monroy and
C.~Ochoa and P.~Nigsch},
title = {{{Extremal antipodal polygons and polytopes}}},
booktitle = {Mexican Conference on Discrete Mathematics and
Computational Geometry},
pages = {11--20},
year = 2013,
address = {Oaxaca, M{\'e}xico},
category = {3b},
pdf = {/files/publications/geometry/acdfon-eapp-13.pdf},
abstract = {Let $S$ be a set of $2n$ points on a circle such that for
each point $p \in S$ also its antipodal (mirrored with
respect to the circle center) point $p'$ belongs to $S$. A
polygon $P$ of size $n$ is called \emph{antipodal} if it
consists of precisely one point of each antipodal pair
$(p,p')$ of $S$. We provide a complete characterization of
antipodal polygons which maximize (minimize, respectively)
the area among all antipodal polygons of $S$. Based on this
characterization, a simple linear time algorithm is
presented for computing extremal antipodal polygons.
Moreover, for the generalization of antipodal polygons to
higher dimensions we show that a similar characterization
does not exist.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ah-fmmrr-93,
author = {O. Aichholzer and H. Hassler},
title = {{{A fast method for modulus reduction in Residue Number
System}}},
booktitle = {Proc. epp'93},
pages = {41--54},
year = 1993,
address = {Vienna, Austria},
category = {3b},
oaich_label = 1,
note = {[IIG-Report-Series 312, TU Graz, Austria, 1991]},
postscript = {/files/publications/geometry/ah-fmmrr-93.ps.gz},
abstract = {Over the last three decades there has been considerable
interest in the implementation of digital computer elements
using hardware based on the residue number system. We
propose a technique to compute a residue in this number
system using a parallel network. Our technique enables
scaling, to. We improve a former result of $O(n)$ cycles to
$O(\log n)$, where $n$ is the number of moduli. The
hardware expense is the same, $O(n^2)$. Further advantages
are that scaling factors can be chosen almost freely
allowing scaling with radix $2$. Negative numbers are
covered as well, requiring no additional effort.
Applications are RSA encryption and scaling.},
originalfile = {/geometry/cggg.bib}
}
@article{aacks-ssmat-13,
author = {J.~Abhau and O.~Aichholzer and S.~Colutto and
B.~Kornberger and O.~Scherzer},
title = {{{Shape Spaces via Medial Axis Transforms for Segmentation
of Complex Geometry in 3{D} Voxel Data}}},
year = 2013,
volume = {7},
number = {1},
journal = {Inverse Problems and Imaging},
pages = {1--25},
note = {},
category = {3a},
abstract = {In this paper we construct a shape space of medial ball
representations from given shape training data using
methods of Computational Geometry and Statistics. The
ultimate goal is to employ the shape space as prior
information in supervised segmentation algorithms for
complex geometries in 3D voxel data. For this purpose, a
novel representation of the shape space (i.e., medial ball
representation) is worked out and its implications on the
whole segmentation pipeline are studied. Such algorithms
have wide applications for industrial processes and medical
imaging, when data are recorded under varying illumination
conditions, are corrupted with high noise or are
occluded.},
originalfile = {/geometry/cggg.bib}
}
@incollection{aa-ssgpf-98,
author = {O. Aichholzer and F. Aurenhammer},
title = {{{Straight skeletons for general polygonal figures in the
plane}}},
booktitle = {Voronoi's Impact on Modern Sciences II},
pages = {7--21},
publisher = {Proc. Institute of Mathematics of the National Academy of
Sciences of Ukraine},
year = 1998,
editor = {A.M. Samoilenko},
volume = 21,
address = {Kiev, Ukraine},
category = 2,
oaich_label = {10a},
postscript = {/files/publications/geometry/aa-ssgpf-98.ps.gz},
abstract = {A novel type of skeleton for general polygonal figures,
the straight skeleton $S(G)$ of a planar straight line
graph $G$, is introduced and discussed. Exact bounds on the
size of $S(G)$ are derived. The straight line structure of
$S(G)$ and its lower combinatorial complexity may make
$S(G)$ preferable to the widely used Voronoi diagram (or
medial axis) of $G$ in several applications. We explain why
$S(G)$ has no Voronoi diagram based interpretation and why
standard construction techniques fail to work. A simple
$O(n)$ space algorithm for constructing $S(G)$ is proposed.
The worst-case running time is $O(n^3 \log n)$, but the
algorithm can be expected to be practically efficient, and
it is easy to implement. We also show that the concept of
$S(G)$ is flexible enough to allow an individual weighting
of the edges and vertices of $G$, without changes in the
maximal size of $S(G)$, or in the method of construction.
Apart from offering an alternative to Voronoi-type
skeletons, these generalizations of $S(G)$ have
applications to the reconstruction of a geographical
terrain from a given river map, and to the construction of
a polygonal roof above a given layout of ground walls.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aah-fps-01a,
author = {O. Aichholzer and L.S. Alboul and F. Hurtado},
title = {{{On Flips in Polyhedral Surfaces}}},
booktitle = {Proc. $17^{th}$ European Workshop on Computational
Geometry CG '2001},
pages = {27--30},
year = 2001,
address = {Berlin, Germany},
category = {3b},
oaich_label = {29},
postscript = {/files/publications/geometry/aah-fps-01a.ps.gz},
htmlnote = {See also our interactive
web-page.},
abstract = {Let $V$ be a finite point set in 3D-space, and let ${\cal
S}(V)$ be the set of triangulated polyhedral surfaces
homeomorphic to a sphere and with vertex set $V$. Let $abc$
and $cbd$ be two adjacent triangles belonging to a surface
$S\in {\cal S}(V)$; the {\sl flip} of the edge $bc$ would
replace these two triangles by the triangles $abd$ and
$adc$. The flip operation is only considered when it does
not produce a self--intersecting surface. In this paper we
show that given two surfaces $S_1, S_2\in {\cal S}(V)$, it
is possible that there is no sequence of flips transforming
$S_1$ into $S_2$, even in the case that $V$ consists of
points in convex position.},
originalfile = {/geometry/cggg.bib}
}
@phdthesis{a-ccphn-97,
author = {O. Aichholzer},
title = {{{Combinatorial \& Computational Properties of the Hypercube
- New Results on Covering, Slicing, Clustering and
Searching on the Hypercube}}},
school = {IGI-TU Graz, Austria},
year = 1997,
category = {5},
oaich_label = {19},
postscript = {/files/publications/geometry/a-ccphn-97.ps.gz},
abstract = { The central topic of this thesis is the $d$-dimensional
hypercube ($d$-cube). Despite of its simple definition, the
$d$-cube has been an object of study from various points of
view. The contributions of this thesis are twofold. In the
combinatorial part we investigate the structure of
hyperplanes intersecting the $d$-cube: How many and which
types of hyperplanes can be spanned by vertices of the
$d$-cube? What is the minimum number of skew hyperplanes
that cover all the vertices of the $d$-cube? What is the
best arrangement of slicing hyperplanes to linearly
separate all neighbours on the $d$-cube? Such results are
then used e.g. for determining the maximal number of facets
a 5-dimensional $0/1$-polytope can achieve. In the second
part of the thesis we consider algorithmical properties of
the $d$-cube. We first obtain efficient clustering methods
for objects represented as binary strings of fixed length
$d$, including various agglomerative hierarchical methods
like single linkage and complete linkage. Utilizing these
hierarchical structures we derive a new efficient approach
to the ${0,1}$-string searching problem, where for a given
set of binary strings of fixed length $d$ and a query
string one asks for the most similar string in this set.
Motivation for investigating these problems stems, among
other areas, from coding theory, communication theory, and
learning theory.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aahkprsv-rsrso-09,
author = {O. Aichholzer and F. Aurenhammer and B. Kornberger and S.
Plantinga and G. Rote and A. Sturm and G. Vegter},
title = {{{Recovering Structure from r-Sampled Objects}}},
booktitle = {Eurographics Symposium on Geometry Processing, special
issue of Computer Graphics Forum 28(5)},
pages = {1349--1360},
pdf = {/files/publications/geometry/aahkprsv-rsrso-09.pdf},
oaich_label = {74b},
year = 2009,
address = {Berlin, Germany},
abstract = {For a surface~$F$ in 3-space that is represented by a
set~$S$ of sample points, we construct a coarse
approximating polytope $P$ that uses a subset of~$S$ as its
vertices and preserves the topology of~$F$. In contrast to
surface reconstruction we do not use all the sample points,
but we try to use as few points as possible. Such a
polytope~$P$ is useful as a `seed polytope' for starting an
incremental refinement procedure to generate better and
better approximations of $F$ based on interpolating
subdivision surfaces or e.g. B\'ezier patches.
Our algorithm starts from an \mbox{$r$-sample} $S$ of $F$.
Based on $S$, a set of surface covering balls with maximal
radii is calculated such that the topology is retained.
From the weighted $\alpha$-shape of a proper subset of
these highly overlapping surface balls we get the desired
polytope. As there is a rather large range for the possible
radii for the surface balls, the method can be used to
construct triangular surfaces from point clouds in a
scalable manner. We also briefly sketch how to combine
parts of our algorithm with existing medial axis algorithms
for balls, in order to compute stable medial axis
approximations with scalable level of detail.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{abdhkkrsu-ggt-03,
author = {O. Aichholzer and D. Bremner and E.D. Demaine and F.
Hurtado and E. Kranakis and H. Krasser and S. Ramaswami and
S. Sethia and J. Urrutia},
title = {{{Geometric Games on Triangulations}}},
booktitle = {Proc. $19^{th}$ European Workshop on Computational
Geometry CG '03 Bonn },
pages = {89--92},
year = 2003,
address = {Bonn, Germany},
category = {3b},
oaich_label = {47},
postscript = {/files/publications/geometry/abdhkkrsu-ggt-03.ps.gz},
abstract = {We analyze several perfect-information combinatorial games
played on planar triangulations. We describe main broad
categories of these games and provide in various situations
polynomial-time algorithms to determine who wins a given
game under optimal play, and ideally, to find a winning
strategy. Relations to relevant existing combinatorial
games, such as Kayles, are also shown.},
originalfile = {/geometry/cggg.bib}
}
@article{acddemoprt-fp-00b,
author = {O. Aichholzer and C. Cort\'{e}s and E.D. Demaine and V.
Dujmovi\'{c} and J. Erickson and H. Meijer and M. Overmars
and B. Palop and S. Ramaswami and G.T. Toussaint},
title = {{{Flipturning Polygons}}},
journal = {Discrete \& Computational Geometry},
year = 2002,
pages = {231--253},
volume = {28},
number = 2,
note = {[Report UU-CS-2000-31, Universiteit Utrecht, The
Netherlands, 2000]},
category = {3a},
oaich_label = {27b},
postscript = {/files/publications/geometry/acddemoprt-fp-00b.ps.gz},
abstract = {A flipturn is an operation that transforms a nonconvex
simple polygon into another simple polygon, by rotating a
concavity 180 degrees around the midpoint of its bounding
convex hull edge. Joss and Shannon proved in 1973 that a
sequence of flipturns eventually transforms any simple
polygon into a convex polygon. This paper describes several
new results about such flipturn sequences. We show that any
orthogonal polygon is convexified after at most $n-5$
arbitrary flipturns, or at most $5(n-4)/6$ well-chosen
flipturns, improving the previously best upper bound of
$(n-1)!/2$. We also show that any simple polygon can be
convexified by at most $n^2-4n+1$ flipturns, generalizing
earlier results of Ahn et al. These bounds depend
critically on how degenerate cases are handled; we
carefully explore several possibilities. We describe how to
maintain both a simple polygon and its convex hull in
$O(\log^4 n)$ time per flipturn, using a data structure of
size $O(n)$. We show that although flipturn sequences for
the same polygon can have very different lengths, the shape
and position of the final convex polygon is the same for
all sequences and can be computed in $O(n \log n)$ time.
Finally, we demonstrate that finding the longest
convexifying flipturn sequence of a simple polygon is
NP-hard.},
originalfile = {/geometry/cggg.bib}
}
@mastersthesis{a-hhkq-92,
author = {O. Aichholzer},
title = {{{{H}yperebenen in {H}yperkuben - {E}ine {K}lassifizierung
und {Q}uantifizierung}}},
school = {IGI-TU Graz, Austria},
year = 1992,
category = {11},
oaich_label = {2},
postscript = {/files/publications/geometry/a-hhkq.ps.gz},
abstract = {In dieser Arbeit werden affine Hyperebenen in
h{\"o}herdimensionalen R{\"a}umen behandelt, die den
$n$-dimensionalen Hyperkubus schneiden. Dabei konzentriert
sich die Untersuchung auf folgende Fragestellung: Wieviele
und welche Arten von Hyperebenen gibt es, die durch
Eckpunkte des Hyperkubus eindeutig festgelegt sind? Neben
der Betrachtung, wieviele solcher Hyperebenen existieren,
wird eine Klassifizierung nach verschiedenen Kriterien wie
Symmetrie, Parallelit{\"a}t zu den Koordinatenachsen,
Anzahl der geschnittenen Eckpunkte etc. untersucht. Die
Arbeit enth{\"a}lt sowohl eine vollst{\"a}ndige enumerative
Berechnung aller relevanten Werte bis einschlie{\ss}lich
der achten Dimension, als auch die theoretische Herleitung
allgemein g{\"u}ltiger S{\"a}tze {\"u}ber solche
Hyperebenen. Die Beitr{\"a}ge dieser Arbeit fallen in das
Gebiet der geometrischen Kombinatorik und finden sowohl in
der Codierungs- und Lerntheorie als auch in der linearen
Optimierung sowie im VLSI-Design Anwendung.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aar-msrp-94a,
author = {O. Aichholzer and H. Alt and G. Rote},
title = {{{Matching Shapes with a Reference Point}}},
booktitle = {Proc. $10^{th}$ Ann. ACM Symp. Computational Geometry},
pages = {85--92},
year = 1994,
address = {Stony Brook, New York, USA},
category = {3b},
oaich_label = {4},
postscript = {/files/publications/geometry/aar-msrp-94.ps.gz},
abstract = {For two given point sets, we present a very simple (almost
trivial) algorithm to translate one set so that the
Hausdorff distance between the two sets is not larger than
a constant factor times the minimum Hausdorff distance
which can be achieved in this way. The algorithm just
matches the so-called Steiner points of the two sets.\\ The
focus of our paper is the general study of reference points
(like the Steiner point) and their properties with respect
to shape matching.\\ For more general transformations than
just translations, our method eliminates several degrees of
freedom from the problem and thus yields good matchings
with improved time bounds.},
originalfile = {/geometry/cggg.bib}
}
@article{aah-fps-01b,
author = {O. Aichholzer and L.S. Alboul and F. Hurtado},
title = {{{On Flips in Polyhedral Surfaces}}},
journal = {International Journal of Foundations of Computer Science
(IJFCS), special issue on Volume and Surface
Triangulations},
year = 2002,
volume = 13,
number = 2,
pages = {303--311},
category = {3a},
oaich_label = {29b},
postscript = {/files/publications/geometry/aah-fps-01b.ps.gz},
abstract = {Let $V$ be a finite point set in 3-space, and let ${\cal
S}(V)$ be the set of triangulated polyhedral surfaces
homeomorphic to a sphere and with vertex set $V$. Let $abc$
and $cbd$ be two adjacent triangles belonging to a surface
$S\in {\cal S}(V)$; the {\sl flip} of the edge $bc$ would
replace these two triangles by the triangles $abd$ and
$adc$. The flip operation is only considered when it does
not produce a self--intersecting surface. In this paper we
show that given two surfaces $S_1, S_2\in {\cal S}(V)$, it
is possible that there is no sequence of flips transforming
$S_1$ into $S_2$, even in the case that $V$ consists of
points in convex position.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aak-ctps-01,
author = {O. Aichholzer and F. Aurenhammer and H. Krasser},
title = {{{On Compatible Triangulations of Point Sets}}},
booktitle = {Proc. $17^{th}$ European Workshop on Computational
Geometry CG '2001},
pages = {23--26},
year = 2001,
address = {Berlin, Germany},
category = {3b},
oaich_label = {28},
postscript = {/files/publications/geometry/aak-ctps-01.ps.gz},
abstract = {Two conjectures on compatible triangulations for planar
point sets are stated and proven for small sets and for
special sets of arbitrary size.},
originalfile = {/geometry/cggg.bib}
}
@article{aak-iubrp-08,
author = {E. Ackerman and O. Aichholzer and B. Keszegh},
title = {{{Improved Upper Bounds on the Reflexivity of Point Sets}}},
year = 2009,
journal = {Computational Geometry: Theory and Applications},
volume = {42},
pages = {241--249},
category = {3a},
oaich_label = {70b},
pdf = {/files/publications/geometry/aak-iubrp-08.pdf},
abstract = {Given a set $S$ of $n$ points in the plane, the
\emph{reflexivity} of $S$, $\rho(S)$, is the minimum number
of reflex vertices in a simple polygonalization of $S$.
Arkin et al. proved that $\rho(S) \le n/2$ for any set $S$,
and conjectured that the tight upper bound is $n/4$. We
show that the reflexivity of any set of $n$ points is at
most $\frac{3}{7}n + O(1) \approx 0.4286n$. Using
computer-aided abstract order type extension the upper
bound can be further improved to $\frac{5}{12}n + O(1)
\approx 0.4167n$.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{abdhkkrsu-pwt-03,
author = {O. Aichholzer and D. Bremner and E.D. Demaine and F.
Hurtado and E. Kranakis and H. Krasser and S. Ramaswami and
S. Sethia and J. Urrutia},
title = {{{Playing with Triangulations}}},
booktitle = {Lecture Notes in Computer Science 2866, Japanese
Conference, JCDCG 2002},
pages = {22--37},
year = 2003,
category = {3b},
oaich_label = {43b},
postscript = {/files/publications/geometry/abdhkkrsu-pwt-03.ps.gz},
abstract = {We analyze several perfect-information combinatorial games
played on planar triangulations. We introduce three broad
categories of such games constructing, transforming and
marking triangulations. In various situations, we develop
polynomial-time algorithms to determine who wins a given
game under optimal play, and to find a winning strategy.
Along the way we show connections to existing combinatorial
games, such as Kayles.},
originalfile = {/geometry/cggg.bib}
}
@article{aak-eotsp-01a,
author = {O. Aichholzer and F. Aurenhammer and H. Krasser},
title = {{{Enumerating Order Types for Small Point Sets with
Applications}}},
journal = {Order},
pages = {265--281},
volume = 19,
year = {2002},
category = {3a},
oaich_label = {31a},
postscript = {/files/publications/geometry/aak-eotsp-01.ps.gz},
htmlnote = {See also our order
type homepage.},
abstract = {Order types are a means to characterize the combinatorial
properties of a finite point configuration. In particular,
the crossing properties of all straight-line segments
spanned by a planar $n$-point set are reflected by its
order type. We establish a complete and reliable data base
for all possible order types of size $n=10$ or less. The
data base includes a realizing point set for each order
type in small integer grid representation. To our
knowledge, no such project has been carried out before. We
substantiate the usefulness of our data base by applying it
to several problems in computational and combinatorial
geometry. Problems concerning triangulations, simple
polygonalizations, complete geometric graphs, and $k$-sets
are addressed. This list of possible applications is not
meant to be exhaustive. We believe our data base to be of
value to many researchers who wish to examine their
conjectures on small point configurations. },
originalfile = {/geometry/cggg.bib}
}
@article{adehost-rcp-00b,
author = {O. Aichholzer and E.D. Demaine and J. Erickson and F.
Hurtado and M. Overmars and M.A. Soss and G.T. Toussaint},
title = {{{Reconfiguring Convex Polygons}}},
journal = {Computational Geometry: Theory and Applications},
year = 2001,
pages = {85--95},
volume = 20,
note = {[Report UU-CS-2000-30, Universiteit Utrecht, The
Netherlands, 2000]},
category = {3a},
oaich_label = {26b},
postscript = {/files/publications/geometry/adehost-rcp-00b.ps.gz},
pdf = {/files/publications/geometry/adehost-rcp-01.pdf},
abstract = {We prove that there is a motion from any convex polygon to
any convex polygon with the same counterclockwise sequence
of edge lengths, that preserves the lengths of the edges,
and keeps the polygon convex at all times. Furthermore, the
motion is ``direct'' (avoiding any intermediate canonical
configuration like a subdivided triangle) in the sense that
each angle changes monotonically throughout the motion. In
contrast, we show that it is impossible to achieve such a
result with each vertex-to-vertex distance changing
monotonically. We also demonstrate that there is a motion
between any two such polygons using three-dimensional moves
known as pivots, although the complexity of the motion
cannot be bounded as a function of the number of vertices
in the polygon.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{abdmss-lpuof-01,
author = {O. Aichholzer and D. Bremner and E.D. Demaine and D.
Meijer and V. Sacrist\'{a}n and M. Soss},
title = {{{Long Proteins with Unique Optimal Foldings in the H-P
Model}}},
booktitle = {Proc. $17^{th}$ European Workshop on Computational
Geometry CG '2001},
pages = {59--62},
year = 2001,
address = {Berlin, Germany},
category = {3b},
oaich_label = {30},
postscript = {/files/publications/geometry/abdmss-lpuof-01.ps.gz},
abstract = {We explore a problem suggested by Brian Hayes in 1998:
what proteins in the two-dimensional
hydrophilic-hydrophobic (H-P) model have {\it unique}
optimal foldings? In particular, we prove that there are
closed chains of monomers (amino acids) with this property
for all (even) lengths; and that there are open monomer
chains with this property fo all lengths divisible by four.
Along the way, we prove and conjecture several results
about bonds in the H-P model.},
originalfile = {/geometry/cggg.bib}
}
@article{aaclp-svd-99,
author = {O. Aichholzer and F. Aurenhammer and D.Z. Chen and D.T.
Lee and E. Papadopoulou},
title = {{{Skew {V}oronoi diagrams}}},
journal = {Int'l. Journal of Computational Geometry \& Applications},
year = 1999,
volume = 9,
pages = {235--247},
category = {3a},
oaich_label = {16a},
postscript = {/files/publications/geometry/aaclp-svd-99.ps.gz},
htmlnote = {Click here for Figures
and Animations of Skew Voronoi Diagrams},
abstract = {On a tilted plane $T$ in three-space, {\em skew
distances\/} are defined as the Euclidean distance plus a
multiple of the signed difference in height. Skew distances
may model realistic environ\-ments more closely than the
Euclidean distance. Voro\-noi diagrams and related problems
under this kind of distances are investigated. A
relationship to convex distance functions and to Euclidean
Voronoi diagrams for planar circles is shown, and is
exploited for a geometric analysis and a plane-sweep
construction of Voronoi diagrams on $T$. An
output-sensitive algorithm running in time $O(n \log h)$ is
developed, where $n$ and $h$ is the number of sites and
non-empty Voronoi regions, respectively. The all nearest
neighbors problem for skew distances, which has certain
features different from its Euclidean counterpart, is
solved in $O(n \log n)$ time.},
originalfile = {/geometry/cggg.bib}
}
@techreport{a-ch-96,
author = {O. Aichholzer},
title = {{{Clustering the Hypercube}}},
institution = {SFB 'Optimierung und Kontrolle', TU Graz, Austria},
year = 1996,
type = {SFB-Report},
number = {F003-93},
category = {5},
oaich_label = {14},
postscript = {/files/publications/geometry/a-ch-96.ps.gz},
abstract = {In this paper we consider various clustering methods for
objects represented as binary strings of fixed length $d$.
The dissimilarity of two given objects is the number of
disagreeing bits, that is, their Hamming distance.
Clustering these objects can be seen as clustering a subset
of the vertices of a $d$-dimensional hypercube, and thus is
a geometric problem in d dimensions. We give algorithms for
various agglomerative hierarchical methods (including
single linkage and complete linkage) as well as for
two-clusterings and divisive methods.\\ We only present
linear space algorithms since for most practical
applications the number of objects to be clustered is
usually to large for non-linear space solutions to be
practicable. All algorithms are easy to implement and the
constants in their asymptotic runtime are small. We give
experimental results for all cluster methods considered,
and for uniformly distributed hypercube vertices as well as
for specially chosen sets. These experiments indicate that
our algorithms work well in practice.},
originalfile = {/geometry/cggg.bib}
}
@article{aafrs-sdccn-13,
author = {B.M.~\'{A}brego and O.~Aichholzer and
S.~Fern\'{a}ndez-Merchant and P.~Ramos and G.~Salazar},
title = {{{Shellable drawings and the cylindrical crossing number of
$K_n$}}},
year = 2014,
journal = {Discrete and Computational Geometry},
volume = {52},
pages = {743--753},
pdf = {/files/publications/geometry/aafrs-sdccn-14.pdf},
doi = {10.1007/s00454-014-9635-0},
eprint = {1309.3665},
archiveprefix = {arXiv},
abstract = {The Harary-Hill Conjecture States that the number of
crossings in any drawing of the complete graph $K_n$ in the
plane is at least $Z(n):=\frac{1}{4}\left\lfloor
\frac{n}{2}\right\rfloor
\left\lfloor\frac{n-1}{2}\right\rfloor \left\lfloor
\frac{n-2}{2}\right\rfloor\left\lfloor
\frac{n-3}{2}\right\rfloor$. In this paper, we settle the
Harary-Hill conjecture for {\em shellable drawings}. We say
that a drawing $D$ of $ K_n $ is {\em $ s $-shellable} if
there exist a subset $ S = \{v_1,v_2,\ldots,v_ s\}$ of the
vertices and a region $R$ of $D$ with the following
property: For all $1 \leq i < j \leq s$, if $D_{ij}$ is the
drawing obtained from $D$ by removing $v_1,v_2,\ldots
v_{i-1},v_{j+1},\ldots,v_{s}$, then $v_i$ and $v_j$ are on
the boundary of the region of $D_{ij}$ that contains $R$.
For $ s\geq n/2 $, we prove that the number of crossings of
any $ s $-shellable drawing of $ K_n $ is at least the
long-conjectured value Z(n). Furthermore, we prove that all
cylindrical, $ x $-bounded, monotone, and 2-page drawings
of $ K_n $ are $ s $-shellable for some $ s\geq n/2 $ and
thus they all have at least $ Z(n) $ crossings. The
techniques developed provide a unified proof of the
Harary-Hill conjecture for these classes of drawings.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{acflsu-cmpwm-11,
author = {O. Aichholzer and M. Cetina and R.~Fabila-Monroy and
J.~Lea{\~n}os and G. Salazar and J.~Urrutia},
title = {{{Convexifying monotone polygons while maintaining internal
visibility}}},
booktitle = {Proc. XIV Encuentros de Geometr\'{\i}a Computacional},
category = {3b},
pages = {35--38},
pdf = {/files/publications/geometry/acflsu-cmpwm-11.pdf},
year = 2011,
address = {Alcal\'a, Spain},
abstract = {Let $P$ be a simple polygon on the plane. Two vertices of
$P$ are visible if the open line segment joining them is
contained in the interior of $P$. In this paper we study
the following questions posed by Devadoss: (1) Is it true
that every non-convex simple polygon has a vertex that can
be continuously moved such that during the process no
vertex-vertex visibility is lost and some vertex-vertex
visibility is gained? (2) Can every simple polygon be
convexified by continuously moving only one vertex at a
time without losing any internal vertex-vertex visibility
during the process? We provide a counterexample to (1). We
note that our counterexample uses a monotone polygon. We
also show that question (2) has a positive answer for
monotone polygons.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{a-e0ssb-98,
author = {O. Aichholzer},
title = {{{Efficient $\{0,1\}$-String Searching Based on
Pre-clustering}}},
booktitle = {Proc. $14^{th}$ European Workshop on Computational
Geometry CG '98},
pages = {11--13},
year = 1998,
address = {Barcelona, Spain},
category = {3b},
oaich_label = {15},
note = {[SFB Report F003-94, TU Graz, Austria, 1996]},
postscript = {/files/publications/geometry/a-e0ssb-98.ps.gz},
abstract = {In this paper we consider the ${0,1}$-string searching
problem. For a given set $S$ of binary strings of fixed
length $d$ and a query string $q$ one asks for the most
similar string in $S$. Thereby the dissimilarity of two
given strings is the number of disagreeing bits, that is,
their Hamming distance. We present an efficient
${0,1}$-string searching algorithm based on hierarchical
pre-clustering. To this end we give several useful
observations on the inter- and intra-cluster distances.\\
The presented algorithms are easy to implement and we give
exhaustive experimental results for uniformly distributed
sets as well as for specially chosen strings. These
experiments indicate that our algorithms work well in
practice.},
originalfile = {/geometry/cggg.bib}
}
@article{aak-cncg-05,
author = {O. Aichholzer and F. Aurenhammer and H. Krasser},
title = {{{On the Crossing Number of Complete Graphs}}},
year = 2006,
journal = {Computing},
volume = 76,
pages = {165--176},
category = {3a},
oaich_label = {36d},
postscript = {/files/publications/geometry/aak-cncg-05.ps.gz},
abstract = { Let $\overline{cr}(G)$ denote the rectilinear crossing
number of a graph~$G$. We determine
$\overline{cr}(K_{11})=102$ and
$\overline{cr}(K_{12})=153$. Despite the remarkable hunt
for crossing numbers of the complete graph~$K_n$ --
initiated by R.~Guy in the 1960s -- these quantities have
been unknown for \mbox{$n>10$} to~date. Our solution mainly
relies on a tailor-made method for enumerating all
inequivalent sets of points (order types) of size~$11$.
Based on these findings, we establish a new upper bound on
$\overline{cr}(K_{n})$ for general~$n$. The bound stems
from a novel construction of drawings of $K_{n}$ with few
crossings.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aafrs-2pcn-12,
author = {Bernardo M.~\'{A}brego and Oswin Aichholzer and Silvia
Fern\'{a}ndez-Merchant and Pedro Ramos and Gelasio Salazar },
title = {{{The 2-page crossing number of $K_{n}$}}},
booktitle = {$28^{th}$ Ann. ACM Symp. Computational Geometry},
category = {3b},
pages = {397--403},
pdf = {/files/publications/geometry/aafrs-pcn-12.pdf},
year = 2012,
address = {Chapel Hill, NC, USA},
abstract = {Around 1958, Hill conjectured that the crossing number
$cr(K_n)$ of the complete graph $K_{n}$ is $$ Z\left(
n\right):=\frac{1}{4}\left\lfloor \frac{n}{2}\right\rfloor
\left\lfloor\frac{n-1}{2}\right\rfloor \left\lfloor
\frac{n-2}{2}\right\rfloor\left\lfloor
\frac{n-3}{2}\right\rfloor $$ and provided drawings of
$K_{n}$ with exactly $Z(n)$ crossings. Towards the end of
the century, substantially different drawings of $K_{n}$
with $Z(n)$ crossings were found. These drawings are
\emph{2-page book drawings}, that is, drawings where all
the vertices are on a line $\ell$ (the spine) and each edge
is fully contained in one of the two half-planes (pages)
defined by~$\ell$. The \emph{2-page crossing number }of
$K_{n} $, denoted by $\nu_{2}(K_n)$, is the minimum number
of crossings determined by a 2-page book drawing of
$K_{n}$. Since $cr(K_n) \le \nu_2(K_n)$ and $\nu_2(K_n) \le
Z(n)$, a natural step towards Hill's Conjecture is the
(formally) weaker conjecture $\nu_2(K_n) = Z(n)$, that was
popularized by Vrt'o. In this paper we develop a novel and
innovative technique to investigate crossings in drawings
of $K_n$, and use it to prove that $\nu_{2}(K_n) = Z(n)$.
To this end, we extend the inherent geometric definition of
$k$-edges for finite sets of points in the plane to
topological drawings of $K_{n}$. We also introduce the
concept of ${\leq}{\leq}k$-edges as a useful generalization
of ${\leq}k$-edges. Finally, we extend a powerful theorem
that expresses the number of crossings in a rectilinear
drawing of $K_{n}$ in terms of its number of $k$-edges to the topological setting.},
originalfile = {/geometry/cggg.bib}
}
@article{aah-nrms-99,
author = {O. Aichholzer and F. Aurenhammer and R. Hainz},
title = {{{New results on {MWT} subgraphs}}},
journal = {Information Processing Letters},
year = 1999,
volume = 69,
pages = {215--219},
category = {3a},
oaich_label = {22},
note = {[SFB Report F003-140, TU Graz, Austria, 1998]},
postscript = {/files/publications/geometry/aah-nrms-99.ps.gz},
abstract = {Let $P$ be a polygon in the plane. We disprove the
conjecture that the so-called LMT-skeleton coincides with
the intersection of all locally minimal triangulations,
$LMT(P)$, even for convex polygons $P$. We introduce an
improved LMT-skeleton algorithm which, for any simple
polygon $P$, exactly computes $LMT(P)$, and thus a larger
subgraph of the minimum-weight triangulation $MWT(P)$. The
algorithm achieves the same in the general point set case
provided the connectedness of the improved LMT-skeleton,
which is given in allmost all practical instances. We
further observe that the $\beta$-skeleton of $P$ is a
subset of $MWT(P)$ for all values $\beta >
\sqrt{\frac{4}{3}}$ provided $P$ is convex or near-convex.
This gives evidence for the tightness of this bound in the
general point set case.},
originalfile = {/geometry/cggg.bib}
}
@incollection{acflsu-cmpwm-12,
author = {O. Aichholzer and M. Cetina and R.~Fabila-Monroy and
J.~Lea{\~n}os and G. Salazar and J.~Urrutia},
title = {{{Convexifying monotone polygons while maintaining internal
visibility}}},
volume = {7579},
issue = {},
editor = {A.~Marquez and P.~Ramos and J.~Urrutia},
booktitle = {Special issue: XIV Encuentros de Geometr\'{\i}a
Computacional ECG2011},
series = {Lecture Notes in Computer Science (LNCS)},
category = {3a},
pages = {98--108},
pdf = {/files/publications/geometry/acflsu-cmpwm-11.pdf},
year = 2012,
publisher = {Springer},
abstract = {Let $P$ be a simple polygon on the plane. Two vertices of
$P$ are visible if the open line segment joining them is
contained in the interior of $P$. In this paper we study
the following questions posed by Devadoss: (1) Is it true
that every non-convex simple polygon has a vertex that can
be continuously moved such that during the process no
vertex-vertex visibility is lost and some vertex-vertex
visibility is gained? (2) Can every simple polygon be
convexified by continuously moving only one vertex at a
time without losing any internal vertex-vertex visibility
during the process? We provide a counterexample to (1). We
note that our counterexample uses a monotone polygon. We
also show that question (2) has a positive answer for
monotone polygons.},
originalfile = {/geometry/cggg.bib}
}
@article{aoss-nptcp-07,
author = {O. Aichholzer and D. Orden and F. Santos and B.
Speckmann},
title = {{{On the Number of Pseudo-Triangulations of Certain Point
Sets}}},
journal = {Journal of Combinatorial Theory, Series A},
year = 2008,
volume = {115(2)},
pages = {254-278},
category = {3a},
oaich_label = {49c},
pdf = {/files/publications/geometry/aoss-nptcp-07.pdf},
abstract = {We pose a monotonicity conjecture on the number of
pseudo-triangulations of any planar point set, and check it
on two prominent families of point sets, namely the
so-called double circle and double chain. The latter has
asymptotically $12^n n^{\Theta(1)}$ pointed
pseudo-triangulations, which lies significantly above the
maximum number of triangulations in a planar point set
known so far.},
originalfile = {/geometry/cggg.bib}
}
@incollection{a-ep0pd-00,
author = {O. Aichholzer},
title = {{{Extremal Properties of 0/1-Polytopes of Dimension 5}}},
booktitle = {Polytopes - Combinatorics and Computation},
publisher = {Birkh{\"a}user},
year = 2000,
editor = {G. Ziegler and G. Kalai},
pages = {111--130},
note = {[SFB-Report F003-132, TU Graz, Austria, 1998]},
category = {2},
oaich_label = {21},
postscript = {/files/publications/geometry/a-ep0pd-00.ps.gz},
htmlnote = {You can also investigate
0/1-polytopes by e-mail!},
abstract = {In this paper we consider polytopes whose vertex
coordinates are $0$ or $1$, so called $0/1$-polytopes. For
the first time we give a complete enumeration of all
$0/1$-polytopes of dimension $5$, which enables us to
investigate various of their combinatorial extremal
properties.\\ For example we show that the maximum number
of facets of a five-dimensional $0/1$-polytope is $40$,
answering an open question of Ziegler. Based on the
complete enumeration for dimension $5$ we obtain new
results for $2$-neighbourly $0/1$-polytopes for higher
dimensions.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aak-iubrp-07,
author = {E. Ackerman and O. Aichholzer and B. Keszegh},
title = {{{Improved Upper Bounds on the Reflexivity of Point Sets}}},
booktitle = {Proc. $19th$ Annual Canadian Conference on Computational
Geometry CCCG 2007},
pages = {29--32},
year = 2007,
address = {Ottawa, Ontario, Canada},
category = {3b},
oaich_label = {70},
pdf = {/files/publications/geometry/aak-iubrp-07.pdf},
abstract = {Given a set $S$ of $n$ points in the plane, the
\emph{reflexivity} of $S$, $\rho(S)$, is the minimum number
of reflex vertices in a simple polygonalization of $S$.
Arkin et al. proved that $\rho(S) \le n/2$ for any set $S$,
and conjectured that the tight upper bound is $n/4$. We
show that the reflexivity of any set of $n$ points is at
most $\frac{3}{7}n + O(1) \approx 0.4286n$. Using
computer-aided abstract order type extension the upper
bound can be further improved to $\frac{5}{12}n + O(1)
\approx 0.4167n$.},
originalfile = {/geometry/cggg.bib}
}
@techreport{aak-prcn-01,
author = {O. Aichholzer and F. Aurenhammer and H. Krasser},
title = {{{Progress on rectilinear crossing numbers}}},
institution = {IGI-TU Graz, Austria},
year = 2002,
abstract = {Let $\overline{cr}(G)$ denote the rectilinear crossing
number of a graph $G$. We show $\overline{cr}(K_{11})=102$
and $\overline{cr}(K_{12})=153$. Despite the remarkable
hunt for the crossing number of the complete graph $K_n$,
initiated by R. Guy in the 1960s, these quantities have
been unknown for $n>10$ to date. We also establish new
upper and lower bounds on $\overline{cr}(K_{n})$ for $13
\leq n \leq 20$, along with an improved general lower bound
for $\overline{cr}(K_{n})$. The results mainly rely on
recent methods developed by the authors for exhaustively
enumerating all combinatorially inequivalent sets of points
(so-called order types). },
oaich_label = {36},
category = {3a},
htmlnote = {See also our crossing
number homepage.},
postscript = {/files/publications/geometry/aak-prcn-01.ps.gz},
originalfile = {/geometry/cggg.bib}
}
@techreport{adr-sltgt-95,
author = {O. Aichholzer and R.L.S. Drysdale and G. Rote},
title = {{{A Simple Linear Time Greedy Triangulation Algorithm for
Uniformly Distributed Points}}},
institution = {TU Graz, Austria},
year = 1995,
type = {IIG-Report-Series},
number = {408},
category = {5},
oaich_label = {5},
note = {Presented at the Workshop on Computational Geometry, Army
MSI Cornell, Stony Brook, 1994},
postscript = {/files/publications/geometry/adr-sltgt-95.ps.gz},
abstract = {The greedy triangulation (GT) of a set $S$ of $n$ points
in the plane is the triangulation obtained by starting with
the empty set and at each step adding the shortest
compatible edge between two of the points, where a
compatible edge is defined to be an edge that crosses none
of the previously added edges. In this paper we present a
simple, practical algorithm that computes the greedy
triangulation in expected time $O(n)$ and space $O(n)$, for
$n$ points drawn independently from a uniform distribution
over some fixed convex shape.\\ This algorithm is an
improvement of the $O(n \log n)$ algorithm of Dickerson,
Drysdale, McElfresh, and Welzl. It uses their basic
approach, but generates only $O(n)$ plausible greedy edges
instead of $O(n \log n)$. It uses some ideas similar to
those presented in Levcopoulos and Lingas's $O(n)$ expected
time algorithm. Since we use more knowledge about the
structure of a random point set and its greedy
triangulation, our algorithm needs only elementary data
structures and simple bucketing techniques. Thus it is a
good deal simpler to explain and to implement than the
algorithm of Levcopoulos and Lingas. },
originalfile = {/geometry/cggg.bib}
}
@article{auv-b6lsb-13,
author = {Oswin Aichholzer and Jorge Urrutia and Birgit
Vogtenhuber},
title = {{{Balanced 6-holes in linearly separable bichromatic point
sets}}},
journal = {Electronic Notes in Discrete Mathematics},
volume = {44},
pages = {181 - 186},
year = 2013,
category = {3b},
note = {Special issue dedicated to LAGOS2013},
doi = {http://dx.doi.org/10.1016/j.endm.2013.10.028},
url = {http://www.sciencedirect.com/science/article/pii/S1571065313002448},
doi = {http://dx.doi.org/10.1016/j.endm.2013.10.028},
abstract = {We consider an Erd{\H{o}}s type question on $k$-holes (empty
$k$-gons) in bichromatic point sets. For a bichromatic
point set $S = R \cup B$, a balanced $2k$-hole in $S$ is
spanned by $k$ points of $R$ and $k$ points of $B$. We show
that if $R$ and $B$ are linearly separable and $|R| = |B| =
n$, then the number of balanced 6-holes in $S$ is at least
$1/15n^2-\Theta(n)$.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aart-tin-95,
author = {O. Aichholzer and F. Aurenhammer and G. Rote and M.
Taschwer},
title = {{{Triangulations intersect nicely}}},
booktitle = {Proc. $11^{th}$ Ann. ACM Symp. Computational Geometry},
pages = {220--229},
year = 1995,
address = {Vancouver, Canada},
category = {3b},
oaich_label = {6},
postscript = {/files/publications/geometry/aart-tin-95.ps.gz},
abstract = {We prove that two different triangulations of the same
planar point set always intersect in a systematic manner,
concerning both their edges and their triangles. As a
consequence, improved lower bounds on the weight of a
triangulation are obtained by solving an assignment
problem. The new bounds cover the previously known bounds
and can be computed in polynomial time. As a by-product, an
easy-to-recognize class of point sets is exhibited where
the minimum-weight triangulation coincides with the greedy
triangulation.},
originalfile = {/geometry/cggg.bib}
}
@article{aak-pc-02,
author = {O. Aichholzer and F. Aurenhammer and H. Krasser},
title = {{{Points and Combinatorics}}},
journal = {Special Issue on Foundations of Information Processing of
{TELEMATIK}},
pages = {12--17},
volume = 1,
year = {2002},
address = {Graz, Austria},
category = {4a},
oaich_label = {40},
postscript = {/files/publications/geometry/aak-pc-02.ps.gz},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aap-qpssc-02,
author = {O. Aichholzer and F. Aurenhammer and B. Palop},
title = {{{Quickest Paths, Straight Skeletons, and the City {V}oronoi
Diagram}}},
year = 2002,
booktitle = {Proc. $18^{th}$ Ann. ACM Symp. Computational Geometry},
pages = {151--159},
address = {Barcelona, Spain},
category = {3b},
oaich_label = {37},
postscript = {/files/publications/geometry/aap-qpssc-02.ps.gz},
abstract = {The city Voronoi diagram is induced by quickest paths, in
the $L_1$~plane speeded up by an isothetic transportation
network. We investigate the rich geometric and algorithmic
properties of city Voronoi diagrams, and report on their
use in processing quickest-path queries.\\ In doing so, we
revisit the fact that not every Voronoi-type diagram has
interpretations in both the distance model and the
wavefront model. Especially, straight skeletons are a
relevant example where an interpretation in the former
model is lacking. We clarify the relation between these
models, and further draw a connection to the
bisector-defined abstract Voronoi diagram model, with the
particular goal of computing the city Voronoi diagram
efficiently. },
originalfile = {/geometry/cggg.bib}
}
@article{aah-sstft-00,
author = {O. Aichholzer and F. Aurenhammer and F. Hurtado},
title = {{{Sequences of spanning trees and a fixed tree theorem}}},
year = 2002,
journal = {Computational Geometry: Theory and Applications},
volume = {21},
number = {1--2},
pages = {3--20},
note = {Special Issue. [Report MA2-IR-00-00026, Universitat
Polite\'cnica de Catalunya, Barcelona, Spain, 2000]},
category = {3a},
oaich_label = {25},
postscript = {/files/publications/geometry/aah-sstft-00.ps.gz},
htmlnote = {You can also download the nice program MST-Tool
we used to check and visualize some of the presented
results!},
abstract = {Let ${\cal T}_S$ be the set of all crossing-free spanning
trees of a planar $n$-point set $S$. We prove that ${\cal
T}_S$ contains, for each of its members $T$, a
length-decreasing sequence of trees $T_o,\ldots,T_k$ such
that $T_o=T$, $T_k=MST(S)$, $T_i$ does not cross $T_{i-1}$
for $i=1,\ldots,k$, and $k=O(\log n)$. Here $MST(S)$
denotes the Euclidean minimum spanning tree of the point
set $S$. As an implication, the number of length-improving
and planar edge moves needed to transform a tree $T \in
{\cal T}_S$ into $MST(S)$ is only $O(n\log n)$. Moreover,
it is possible to transform any two trees in ${\cal T}_S$
into each other by means of a local and constant-size edge
slide operation. Applications of these results to morphing
of simple polygons are possible by using a crossing-free
spanning tree as a skeleton description of a polygon.},
originalfile = {/geometry/cggg.bib}
}
@article{ahk-twpst-07,
author = {O. Aichholzer and C. Huemer and H. Krasser},
title = {{{Triangulations Without Pointed Spanning Trees}}},
year = 2008,
volume = {40},
number = {1},
journal = {Computational Geometry: Theory and Applications},
pages = {79--83},
category = {3a},
oaich_label = {50b},
pdf = {/files/publications/geometry/ahk-twpst-07.pdf},
abstract = {Problem $50$ in the Open Problems Project~\cite{OPP} asks
whether any triangulation on a point set in the plane
contains a pointed spanning tree as a subgraph. We provide
a counterexample. As a consequence we show that there exist
triangulations which require a linear number of edge flips
to become Hamiltonian. },
originalfile = {/geometry/cggg.bib}
}
@inproceedings{adehost-rcp-00a,
author = {O. Aichholzer and E.D. Demaine and J. Erickson and F.
Hurtado and M. Overmars and M.A. Soss and G.T. Toussaint},
title = {{{Reconfiguring Convex Polygons}}},
booktitle = {Proc. $12th$ Annual Canadian Conference on Computational
Geometry CCCG 2000},
pages = {17--20},
year = 2000,
address = {Fredericton, New Brunswick, Canada},
category = {3b},
oaich_label = {26},
postscript = {/files/publications/geometry/adehost-rcp-00a.ps.gz},
abstract = {We prove that there is a motion from any convex polygon to
any convex polygon with the same counterclockwise sequence
of edge lengths, that preserves the lengths of the edges,
and keeps the polygon convex at all times. Furthermore, the
motion is ``direct'' (avoiding any intermediate canonical
configuration like a subdivided triangle) in the sense that
each angle changes monotonically throughout the motion. In
contrast, we show that it is impossible to achieve such a
result with each vertex-to-vertex distance changing
monotonically.},
originalfile = {/geometry/cggg.bib}
}
@article{aafrs-2pcn-13,
author = {Bernardo M.~\'{A}brego and Oswin Aichholzer and Silvia
Fern\'{a}ndez-Merchant and Pedro Ramos and Gelasio Salazar },
title = {{{The 2-page crossing number of $K_{n}$}}},
journal = {Discrete \& Computational Geometry},
year = 2013,
volume = {49},
category = {3a},
number = {4},
pages = {747-777},
pdf = {/files/publications/geometry/aafrs-pcn-12.pdf},
abstract = {Around 1958, Hill conjectured that the crossing number
$cr(K_n)$ of the complete graph $K_{n}$ is $$ Z\left(
n\right):=\frac{1}{4}\left\lfloor \frac{n}{2}\right\rfloor
\left\lfloor\frac{n-1}{2}\right\rfloor \left\lfloor
\frac{n-2}{2}\right\rfloor\left\lfloor
\frac{n-3}{2}\right\rfloor $$ and provided drawings of
$K_{n}$ with exactly $Z(n)$ crossings. Towards the end of
the century, substantially different drawings of $K_{n}$
with $Z(n)$ crossings were found. These drawings are
\emph{2-page book drawings}, that is, drawings where all
the vertices are on a line $\ell$ (the spine) and each edge
is fully contained in one of the two half-planes (pages)
defined by~$\ell$. The \emph{2-page crossing number }of
$K_{n} $, denoted by $\nu_{2}(K_n)$, is the minimum number
of crossings determined by a 2-page book drawing of
$K_{n}$. Since $cr(K_n) \le \nu_2(K_n)$ and $\nu_2(K_n) \le
Z(n)$, a natural step towards Hill's Conjecture is the
(formally) weaker conjecture $\nu_2(K_n) = Z(n)$, that was
popularized by Vrt'o. In this paper we develop a novel and
innovative technique to investigate crossings in drawings
of $K_n$, and use it to prove that $\nu_{2}(K_n) = Z(n)$.
To this end, we extend the inherent geometric definition of
$k$-edges for finite sets of points in the plane to
topological drawings of $K_{n}$. We also introduce the
concept of ${\leq}{\leq}k$-edges as a useful generalization
of ${\leq}k$-edges. Finally, we extend a powerful theorem
that expresses the number of crossings in a rectilinear
drawing of $K_{n}$ in terms of its number of $k$-edges to the topological setting.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aabkmmr-eshc-01,
author = {O. Aichholzer and F. Aurenhammer and B. Brandtst{\"a}tter
and H. Krasser and C. Magele and M. M{\"u}hlmann and W.
Renhart},
title = {{{Evolution Strategy and Hierarchical Clustering}}},
booktitle = {$13^{th}$ COMPUMAG Conference on the Computation of
Electromagnetic Fields},
year = 2001,
address = {Lyon-Evian, France},
category = {7},
oaich_label = {35},
postscript = {/files/publications/geometry/aabkmmr-eshc-01.ps.gz},
abstract = {Multi-objective optimization problems, in general, exhibit
several local optima besides a global one. A desirable
feature of any optimization strategy would therefore be to
supply the user with as many information as possible about
local optima on the way to the global solution. In this
paper a hierarchical clustering algorithm implemented into
a higher order Evolution Strategy is applied to achieve
these goals.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{a-pt-97,
author = {O. Aichholzer},
title = {{{The Path of a Triangulation}}},
booktitle = {Proc. $13^{th}$ European Workshop on Computational
Geometry CG '97},
pages = {1--3},
year = 1997,
address = {W{\"u}rzburg, Germany},
category = {3b},
oaich_label = {17},
postscript = {/files/publications/geometry/a-pt-97.ps.gz},
htmlnote = {For an implementation see my page on triangulation
counting.},
abstract = {For a planar point set $S$ let $T$ be a triangulation of
$S$ and $l$ a line properly intersecting $T$. We show that
there always exists a unique path in $T$ with certain
properties with respect to $l$. This path is then
generalized to (non triangulated) point sets restricted to
the interior of simple polygons. This so-called
triangulation path enables us to treat several
triangulation problems on planar point sets in a divide \&
conquer-like manner. For example, we give the first
algorithm for counting triangulations of a planar point set
which is observed to run in time sublinear in the number of
triangulations. Moreover, the triangulation path proves to
be useful for the computation of optimal triangulations.},
originalfile = {/geometry/cggg.bib}
}
@article{aahv-gceps-07,
author = {O. Aichholzer and F. Aurenhammer and C. Huemer and B.
Vogtenhuber},
title = {{{Gray code enumeration of plane straight-line graphs}}},
journal = {Graphs and Combinatorics (Springer)},
pages = {467--479},
volume = {23(5)},
category = {3a},
oaich_label = {62b},
year = 2007,
doi = {10.1007/s00373-007-0750-z},
postscript = {/files/publications/geometry/aahv-gceps-07.pdf},
abstract = {We develop Gray code enumeration schemes for geometric
straight-line graphs in the plane. The considered graph
classes include plane graphs, connected plane graphs, and
plane spanning trees. Previous results were restricted to
the case where the underlying vertex set is in convex
position.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aahk-tct-01,
author = {O. Aichholzer and F. Aurenhammer and F. Hurtado and H.
Krasser},
title = {{{Towards Compatible Triangulations}}},
booktitle = {Proc. $7^{th}$ Ann. Int'l. Computing and Combinatorics
Conf. COCOON'01, Lecture Notes in Computer Science},
pages = {101--110},
year = 2001,
volume = {2108},
address = {Guilin, China},
editor = {Jie Wang},
publisher = {Springer Verlag},
category = {3b},
oaich_label = {32},
postscript = {/files/publications/geometry/aahk-tct-01.ps.gz},
abstract = {We state the following conjecture: any two planar
$n$-point sets (that agree on the number of convex hull
points) can be triangulated in a compatible manner, i.e.,
such that the resulting two planar graphs are isomorphic.
The conjecture is proved true for point sets with at most
three interior points. We further exhibit a class of point
sets which can be triangulated compatibly with any other
set (that satisfies the obvious size and hull
restrictions). Finally, we prove that adding a small number
of Steiner points (the number of interior points minus two)
always allows for compatible triangulations.},
originalfile = {/geometry/cggg.bib}
}
@article{aaks-cmpt-03,
author = {O. Aichholzer and F. Aurenhammer and H. Krasser and B.
Speckmann},
title = {{{Convexity Minimizes Pseudo-Triangulations}}},
journal = {Computational Geometry: Theory and Applications},
volume = {28},
number = 1,
pages = {3--10},
year = 2004,
category = {3a},
oaich_label = {42b},
postscript = {/files/publications/geometry/aaks-cmpt-03.ps.gz},
abstract = {The number of minimum pseudo-triangulations is minimized
for point sets in convex position.},
originalfile = {/geometry/cggg.bib}
}
@article{akpv-got-14,
author = {Oswin Aichholzer and Matias Korman and Alexander Pilz and Birgit Vogtenhuber},
title = {{{Geodesic Order Types}}},
journal = {Algorithmica},
volume = {70},
number = {1},
year = {2014},
pages = {112--128},
doi = {http://dx.doi.org/10.1007/s00453-013-9818-8},
eprint = {1708.06064},
archiveprefix = {arXiv},
archiveprefix = {arXiv},
url = {http://link.springer.com/article/10.1007\%2Fs00453-013-9818-8},
pdf = {/files/publications/geometry/akpv-got-14.pdf},
abstract = {The geodesic between two points a and b in the interior of a simple polygon P is the shortest polygonal path inside P that connects a to b. It is thus the natural generalization of straight line segments on unconstrained point sets to polygonal environments. In this paper we use this extension to generalize the concept of the order type of a set of points in the Euclidean plane to geodesic order types. In particular, we show that, for any set S of points and an ordered subset B ⊆ S of at least four points, one can always construct a polygon P such that the points of B define the geodesic hull of S w.r.t. P, in the specified order. Moreover, we show that an abstract order type derived from the dual of the Pappus arrangement can be realized as a geodesic order type.},
originalfile = {/geometry/cggg.bib}
}
@article{aaddfhlsw-cfs-14,
author = {Oswin Aichholzer and Greg Aloupis and Erik D. Demaine and Martin L. Demaine and S{\'a}ndor P. Fekete and Michal Hoffmann and Anna Lubiw and Jack Snoeyink and Andrew Winslow},
title = {{{Covering Folded Shapes}}},
journal = {Journal of Computational Geometry},
year = {2014},
volume = {5},
issue = {1},
pages = {150--168},
url = {http://jocg.org/index.php/jocg/article/view/2930},
abstract = {Can folding a piece of paper flat make it larger? We explore whether a shape
$S$ must be scaled to cover a flat-folded copy of itself. We consider both
single folds and arbitrary folds (continuous piecewise isometries $S\rightarrow
R^2$). The underlying problem is motivated by computational origami, and is
related to other covering and fixturing problems, such as Lebesgue's universal
cover problem and force closure grasps. In addition to considering special
shapes (squares, equilateral triangles, polygons and disks), we give upper and
lower bounds on scale factors for single folds of convex objects and arbitrary
folds of simply connected objects.},
originalfile = {/geometry/cggg.bib}
}
@article{abbbkrtv-t3c-2014,
title = {{{Theta-3 is connected}}},
author = {Oswin Aichholzer and Sang Won Bae and Luis Barba and
Prosenjit Bose and Matias Korman and Andr\'e van Renssen and
Perouz Taslakian and Sander Verdonschot},
journal = {Computational Geometry Theory and Application},
volume = {47},
issue = {9},
year = {2014},
pages = {910--917},
doi = {10.1016/j.comgeo.2014.05.001},
abstract = {In this paper, we show that the $\Theta$-graph with three cones is
connected. We also provide an alternative proof of the connectivity of
the Yao graph with three cones.},
originalfile = {/geometry/cggg.bib}
}
@article{abbbkrtv-t3c-2015,
title = {{{Reprint of: Theta-3 is connected}}},
author = {Oswin Aichholzer and Sang Won Bae and Luis Barba and
Prosenjit Bose and Matias Korman and Andr\'e van Renssen and
Perouz Taslakian and Sander Verdonschot},
journal = {Computational Geometry Theory and Application},
volume = {48},
issue = {5},
year = {2015},
pages = {369--442},
doi = {10.1016/j.comgeo.2015.01.002},
abstract = {In this paper, we show that the $\Theta$-graph with three cones is
connected. We also provide an alternative proof of the connectivity of
the Yao graph with three cones.},
originalfile = {/geometry/cggg.bib}
}
@article{aam-dcgnc-15,
title = {{{Disjoint compatibility graph of non-crossing matchings of points in convex position}}},
author = {Oswin Aichholzer and Andrei Asinowski and Tillmann Miltzow},
journal = {The Electronic Journal of Combinatorics},
volume = {22},
issue = {1},
year = {2015},
pages = {1--65},
url = {http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i1p65},
pdf = {/files/publications/geometry/aam-dcgnc-15.pdf},
abstract = {Let $X_{2k}$ be a set of $2k$ labeled points in convex
position in the plane. We consider geometric
non-intersecting straight-line perfect matchings of
$X_{2k}$. Two such matchings, $M$ and $M'$, are
\textit{disjoint compatible} if they do not have
common edges, and no edge of $M$ crosses an edge of
$M'$. Denote by $\dcm_k$ the graph whose vertices
correspond to such matchings, and two vertices are
adjacent if and only if the corresponding matchings
are disjoint compatible. We show that for each $k
\geq 9$, the connected components of $\dcm_k$ form
exactly three isomorphism classes -- namely, there
is a certain number of isomorphic \textit{small}
components, a certain number of isomorphic
\textit{medium} components, and one \textit{big}
component. The number and the structure of small
and medium components is determined precisely.},
originalfile = {/geometry/cggg.bib}
}
@article{acdfon-ceap-15,
title = {{{Characterization of extremal antipodal polygons}}},
author = {O.~Aichholzer and L.E.~Caraballo and J.M.~D\'{\i}az-B{\'a}{\~n}ez and R.~Fabila-Monroy and C.~Ochoa and P.~Nigsch},
journal = {Graphs and Combinatorics},
volume = {31},
issue = {2},
year = {2015},
pages = {321--333},
doi = {10.1007/s00373-015-1548-z},
pdf = {/files/publications/geometry/acdfon-ceap-15.pdf},
abstract = {Let $S$ be a set of $2n$ points on a circle such that for each point $p \in S$ also its antipodal (mirrored with respect to the circle center) point $p'$ belongs to $S$. A polygon $P$ of size $n$ is called \emph{antipodal} if it consists of precisely one point of each antipodal pair $(p,p')$ of~$S$.
We provide a complete characterization of antipodal polygons which maximize (minimize, respectively) the area among all antipodal polygons of~$S$. Based on this characterization, a simple linear time algorithm is presented for computing extremal antipodal polygons. Moreover, for the generalization of antipodal polygons to higher dimensions we show that a similar characterization does not exist.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aafrv-nsdfc-14,
author = {Bernardo M.~\'{A}brego and Oswin Aichholzer and Silvia Fern\'{a}ndez-Merchant and Pedro Ramos and Birgit
Vogtenhuber},
title = {{{Non-Shellable Drawings of $K_n$ with Few Crossings}}},
booktitle = {Proc. $26^{th}$ Annual Canadian Conference on
Computational Geometry CCCG 2014},
pages = {online},
year = 2014,
address = {Halifax, Nova Scotia, Canada},
category = {3b},
abstract = {In the early 60s, Harary and Hill conjectured
$H(n):=\frac{1}{4}\lfloor\frac{n}{2}\rfloor\lfloor\frac{n-1}{2}\rfloor\lfloor\frac{n-2}{2}\rfloor\lfloor\frac{n-3}{2}\rfloor$
to be the minimum number of crossings among all drawings of the complete graph $K_n$.
It has recently been shown that this conjecture holds for so-called
shellable drawings of $ K_n $. For $ n \geq 11 $ odd, we construct a non-shellable family of
drawings of $ K_n $ with exactly $H(n)$ crossings. In particular,
every edge in our drawings is intersected by at least one other
edge. So far only two other families were known to achieve the
conjectured minimum of crossings, both of them being shellable.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ahhv-ssmvd-14,
author = {Oswin Aichholzer and Thomas Hackl and Stefan Huber and Birgit Vogtenhuber},
title = {{{Straight Skeletons by Means of Voronoi Diagrams Under Polyhedral Distance Functions}}},
booktitle = {Proc. $26^{th}$ Annual Canadian Conference on
Computational Geometry (CCCG 2014)},
pages = {online},
year = 2014,
address = {Halifax, Nova Scotia, Canada},
category = {3b},
arxiv = {},
pdf = {/files/publications/geometry/ahhv-ssmvd-14.pdf},
thackl_label = {42C},
abstract = {We consider the question under which circumstances the straight
skeleton and the Voronoi diagram of a given input shape coincide.
More precisely, we investigate convex distance functions that stem
from centrally symmetric convex polyhedra as unit balls and derive
sufficient and necessary conditions for input shapes in order to
obtain identical straight skeletons and Voronoi diagrams with
respect to this distance function.
This allows us to present a new approach for generalizing straight
skeletons by means of Voronoi diagrams, so that the straight
skeleton changes continuously when vertices of the input shape are
dislocated, that is, no discontinuous changes as in the Euclidean
straight skeleton occur.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ahkr-dred-14,
author = {Oswin Aichholzer and Michael Hoffmann and Marc Van Kreveld and G\"unter Rote},
title = {{{Graph Drawings with Relative Edge Length Specifications}}},
booktitle = {Proc. $26^{th}$ Annual Canadian Conference on
Computational Geometry CCCG 2014},
pages = {online},
year = 2014,
address = {Halifax, Nova Scotia, Canada},
category = {3b},
abstract = { We study plane straight-line embeddings of graphs where certain
edges are specified to be longer than other edges. We analyze which
graphs are universal in the sense that they allow a planar embedding
for any total, strict order on the edge lengths. In addition, we
also briefly consider circular arc drawings with relative edge
length specifications.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ahkklpsw-ppstp-14,
author = {Oswin Aichholzer and Thomas Hackl and Matias Korman and Marc
van Kreveld and Maarten L\"offler and Alexander Pilz and Bettina Speckmann and Emo Welzl},
title = {{{Packing Plane Spanning Trees and Paths in Complete Geometric Graphs}}},
booktitle = {Proc. $26^{th}$ Annual Canadian Conference on
Computational Geometry (CCCG 2014)},
pages = {online},
year = 2014,
address = {Halifax, Nova Scotia, Canada},
category = {3b},
arxiv = {},
pdf = {/files/publications/geometry/ahkklpsw-ppstp-14.pdf},
thackl_label = {43C},
abstract = {We consider the following question: How many
edge-disjoint plane spanning trees are contained in a complete
geometric graph $GK_n$ on any set $S$ of $n$ points in general
position in the plane?},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{afhpruv-otcsl-14,
author = {Oswin Aichholzer and Ruy Fabila-Monroy and Ferran Hurtado
and Pablo Perez-Lantero and Andres J. Ruiz-Vargas and Jorge Urrutia and Birgit Vogtenhuber},
title = {{{Order types and cross-sections of line arrangements in $R^3$}}},
booktitle = {Proc. $26^{th}$ Annual Canadian Conference on
Computational Geometry CCCG 2014},
pages = {online},
year = 2014,
address = {Halifax, Nova Scotia, Canada},
category = {3b},
abstract = {We consider sets of $n$ labeled lines in general position in ${{\sf l} \kern -.10em {\sf R} }^3$, and
study the order types of point sets that stem from the intersections
of the lines with (directed) planes, not parallel to any given line.
As a main result we show that the number of different order
types that can be obtained as cross-sections of these lines is
$O(n^9)$, and that this bound is tight.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{abddefhks-fppc-15,
author = {Oswin~Aichholzer and Michael~Biro and Erik~Demaine and Martin~Demaine and David~Eppstein
and S\'{a}ndor~P.~Fekete and Adam~Hesterberg and Irina~Kostitsyna and Christiane~Schmidt},
title = {{{Folding Polyominoes into (Poly)Cubes}}},
booktitle = {Proc. $27^{th}$ Annual Canadian Conference on
Computational Geometry CCCG 2015},
pages = {101-106},
year = 2015,
address = {Kingston, Ontario, Canada},
url = {http://research.cs.queensu.ca/cccg2015/CCCG15-papers/CCCG'15_Proc.html},
category = {3b},
abstract = {We study the problem of folding a given polyomino $S$ into a polycube $C$ under different folding models,
allowing faces of $C$ to be covered multiple times.},
originalfile = {/geometry/cggg.bib}
}
@article{abddefhks-fppc-18,
author = {Oswin~Aichholzer and Michael~Biro and Erik~Demaine and Martin~Demaine and David~Eppstein
and S\'{a}ndor~P.~Fekete and Adam~Hesterberg and Irina~Kostitsyna and Christiane~Schmidt},
title = {{{Folding Polyominoes into (Poly)Cubes}}},
journal = {International Journal of Computational Geometry \& Applications},
year = 2018,
volume = {28},
number = {3},
pages = {197--226},
doi = {https://doi.org/10.1142/S0218195918500048},
arxiv = {1712.09317},
abstract = {We study the problem of folding a given polyomino $P$ into a polycube~$Q$,
allowing faces of $Q$ to be covered multiple times.
First, we define a variety of folding models according to whether the folds
(a)~must be along grid lines of $P$ or can divide squares in half
(diagonally and/or orthogonally),
(b)~must be mountain or can be both mountain and valley,
(c)~can remain flat (forming an angle of $180^\circ$), and
(d)~whether the folding must lie on just the polycube surface or can
have interior faces as well.
Second, we give all inclusion relations among all models that fold on the
grid lines of~$P$.
Third, we characterize all polyominoes that can fold into a unit cube,
in some models.
Fourth, we give a linear-time dynamic programming algorithm to fold a
tree-shaped polyomino into a constant-size polycube, in some models.
Finally, we consider the triangular version of the problem,
characterizing which polyiamonds fold into a regular tetrahedron.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ahlmv-efdpc-14,
author = {Oswin Aichholzer and Thomas Hackl and Sarah Lutteropp and Tamara Mchedlidze and Birgit Vogtenhuber},
title = {{{Embedding Four-directional Paths on Convex Point Sets}}},
booktitle = {Proc. $22^{nd}$ International Symposium on Graph Drawing (GD 2014)},
series = {Lecture Notes in Computer Science (LNCS)},
volume = {8871},
nopublisher = {Springer, Heidelberg},
editor = {C.~Duncan and A.~Symvonis},
pages = {355--366},
year = 2014,
address = {W{\"u}rzburg, Germany},
category = {3b},
arxiv = {1408.4933},
pdf = {/files/publications/geometry/ahlmv-efdpc-14.pdf},
thackl_label = {44C},
abstract = {A directed path whose edges are assigned labels ``up'', ``down'', ``right'', or ``left'' is called \emph{four-directional}, and \emph{three-directional} if at most three out of the four labels are used.
A \emph{direction-consistent embedding} of an \mbox{$n$-vertex} four-directional path $P$ on a set $S$ of $n$ points in the plane
is a straight-line drawing of $P$ where each vertex of $P$ is mapped to a distinct point of $S$ and every edge points to the direction specified by its label.
We study planar direction-consistent embeddings of three- and four-directional paths and provide a complete picture of the problem for convex point sets.},
originalfile = {/geometry/cggg.bib}
}
@article{ahlmv-efdpc-15,
author = {Oswin Aichholzer and Thomas Hackl and Sarah Lutteropp and Tamara Mchedlidze and Birgit Vogtenhuber},
title = {{{Embedding Four-directional Paths on Convex Point Sets}}},
journal = {Journal of Graph Algorithms and Applications},
year = 2015,
volume = {19},
number = {2},
pages = {743--759},
thackl_label = {44J},
category = {3a},
issn = {1526-1719},
doi = {http://dx.doi.org/10.7155/jgaa.00368},
arxiv = {1408.4933},
pdf = {/files/publications/geometry/ahlmv-efdpc-14.pdf},
abstract = {A directed path whose edges are assigned labels
``up'', ``down'', ``right'', or ``left'' is called
\emph{four-directional}, and
\emph{three-directional} if at most three out of the
four labels are used. A \emph{direction-consistent
embedding} of an \mbox{$n$-vertex} four-directional
path $P$ on a set $S$ of $n$ points in the plane is
a straight-line drawing of $P$ where each vertex of
$P$ is mapped to a distinct point of $S$ and every
edge points to the direction specified by its label.
We study planar direction-consistent embeddings of
three- and four-directional paths and provide a
complete picture of the problem for convex point
sets.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{acklv-rpsot-14,
author = {Oswin Aichholzer and
Jean Cardinal and
Vincent Kusters and
Stefan Langerman and
Pavel Valtr},
title = {{{Reconstructing Point Set Order Types from Radial Orderings}}},
booktitle = {Algorithms and Computation - 25th International Symposium, {ISAAC}
2014, Jeonju, Korea, December 15-17, 2014, Proceedings},
pages = {15--26},
year = {2014},
crossref = {DBLP:conf/isaac/2014},
doi = {10.1007/978-3-319-13075-0_2},
timestamp = {Mon, 10 Nov 2014 13:24:45 +0100},
biburl = {http://dblp.uni-trier.de/rec/bib/conf/isaac/AichholzerCKLV14},
bibsource = {dblp computer science bibliography, http://dblp.org},
pdf = {/files/publications/geometry/acklv-rpsot.pdf},
abstract = {We consider the problem of reconstructing the combinatorial structure of a
set of $n$ points in the plane given partial information on the relative
position of the points. This partial information consists of the radial
ordering, for each of the $n$ points, of the $n-1$ other points around it. We
show that this information is sufficient to reconstruct the chirotope, or
labeled order type, of the point set, provided its convex hull has size at
least four. Otherwise, we show that there can be as many as $n-1$ distinct
chirotopes that are compatible with the partial information, and this bound
is tight. Our proofs yield polynomial-time reconstruction algorithms. These
results provide additional theoretical insights on previously studied
problems related to robot navigation and visibility-based reconstruction.},
originalfile = {/geometry/cggg.bib}
}
@article{acklv-rpsot-16,
author = {Oswin Aichholzer and
Jean Cardinal and
Vincent Kusters and
Stefan Langerman and
Pavel Valtr},
title = {{{Reconstructing Point Set Order Types from Radial Orderings}}},
journal = {International Journal of Computational Geometry \& Applications},
year = 2016,
volume = {26},
number = {3/4},
pages = {167--184},
category = {3a},
doi = {10.1142/S0218195916600037},
pdf = {/files/publications/geometry/acklv-rpsot.pdf},
abstract = {We consider the problem of reconstructing the combinatorial structure of a
set of $n$ points in the plane given partial information on the relative
position of the points. This partial information consists of the radial
ordering, for each of the $n$ points, of the $n-1$ other points around it. We
show that this information is sufficient to reconstruct the chirotope, or
labeled order type, of the point set, provided its convex hull has size at
least four. Otherwise, we show that there can be as many as $n-1$ distinct
chirotopes that are compatible with the partial information, and this bound
is tight. Our proofs yield polynomial-time reconstruction algorithms. These
results provide additional theoretical insights on previously studied
problems related to robot navigation and visibility-based reconstruction.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{bkhas-pnmmh-14,
author = {Sven Bock and Roland Kl{\"o}bl and Thomas Hackl and Oswin Aichholzer and
Gerald Steinbauer},
title = {{"{Playing Nine Men's Morris with the Humanoid Robot Nao}"}},
booktitle = {Proc. Austrian Robotics Workshop (AWS 2014)},
address = {Linz, Austria},
pages = {58--63},
category = {3b},
thackl_label = {45C},
pdf = {/files/publications/geometry/bkhas-pnmmh-14.pdf},
year = {2014},
abstract = {Playing games is an important aspect in human life
in order to develop skills or in terms of
entertainment. Games also play a major role in
research such as Artificial Intelligence and
Robotics. In this paper we present an approach to
enable the humanoid robot Nao to play a board game
against a human opponent. We discuss the challenges
that arise by the task of playing a board game with
a humanoid robot, provide solutions for the Nao, and
introduce our proof-of-concept implementation for
the board game Nine Men's Morris. Finally, we will
present a first experimental evaluation of the
approach. The main contribution of this paper is the
integration of various techniques into one real
robot system, enabling it to manage a complex task
such as playing a board game.},
originalfile = {/geometry/cggg.bib}
}
@article{amp-fdtsp-15,
author = {Oswin Aichholzer and
Wolfgang Mulzer and
Alexander Pilz},
title = {{{Flip Distance Between Triangulations of a Simple Polygon is {NP}-Complete}}},
journal = {Discrete Comput. Geom.},
volume = {54},
number = {2},
pages = {368--389},
year = {2015},
url = {http://dx.doi.org/10.1007/s00454-015-9709-7},
doi = {10.1007/s00454-015-9709-7},
timestamp = {Thu, 23 Jul 2015 10:18:03 +0200},
biburl = {http://dblp.uni-trier.de/rec/bib/journals/dcg/AichholzerMP15},
bibsource = {dblp computer science bibliography, http://dblp.org},
abstract = {Let $T$ be a triangulation of a simple polygon.
A \emph{flip} in~$T$ is the operation of replacing one diagonal of~$T$
by a different one such that the resulting graph is again
a triangulation. The \emph{flip distance} between two triangulations is
the smallest number of flips required to transform one triangulation into the
other. For the special case of convex polygons, the problem of determining the
shortest flip distance between two triangulations is equivalent to determining
the rotation distance between two binary trees, a central problem which is still
open after over 25 years of intensive study.
We show that computing the flip distance between two
triangulations of a simple polygon is NP-hard. This complements a recent
result that shows APX-hardness of determining the flip distance between two
triangulations of a planar point set.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ahpsv-dmgdc-15,
author = {Oswin Aichholzer and Thomas Hackl and Alexander Pilz and Gelasio Salazar and Birgit Vogtenhuber},
title = {{{Deciding monotonicity of good drawings of the complete graph}}},
booktitle = {Proc. XVI Spanish Meeting on Computational Geometry (EGC 2015)},
pages = {33--36},
year = {2015},
category = {3b},
thackl_label = {47C},
pdf = {/files/publications/geometry/ahpsv-dmgdc-15.pdf},
abstract = {We describe an $O(n^5)$ time algorithm for deciding whether a good drawing of the complete graph $K_n$, given in terms of its rotation system, can be re-drawn using only $x$-monotone arcs.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{abhprvv-hitcps-16,
author = {O.~Aichholzer and M.~Balko and T.~Hackl and A.~Pilz and P.~Ramos and B.~Vogtenhuber and P.~Valtr},
title = {{{Holes in two convex point sets}}},
booktitle = {Proc. $32^{st}$ European Workshop on Computational Geometry EuroCG '16},
pages = {263--266},
year = 2016,
address = {Lugano, Switzerland},
category = {3b},
eprint = {},
pdf = {/files/publications/geometry/abhprvv-hitcps-16.pdf},
thackl_label = {52C},
abstract = {Let $S$ be a finite set of $n$ points in the plane in
general position. A $k-hole$ of $S$ is a simple
polygon with $k$ vertices from $S$ and no points of
$S$ in its interior. A simple polygon $P$ is
$l$-convex if no straight line intersects the
interior of $P$ in more than $l$ connected
components. Moreover, a point set $S$ is $l$-convex
if there exists an $l$-convex polygonalization of
$S$. Considering a typical Erd{\H{o}}s-Szekeres type
problem we show that every 2-convex point set of
size $n$ contains a convex hole of size $\Omega(log n)$.
This is in contrast to the well known fact that
there exist general point sets of arbitrary size
that do not contain a convex 7-hole. Further, we
show that our bound is tight by providing a
construction for 2-convex point sets with holes of
size at most $O(log n)$.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{abhprvv-hi2cps-17,
author = {O.~Aichholzer and M.~Balko and T.~Hackl and A.~Pilz and P.~Ramos and P.~Valtr and B.~Vogtenhuber},
title = {{{Holes in 2-convex point sets}}},
booktitle = {Proc. $28^{th}$ International Workshop on Combinatorial Algorithms (IWOCA2017)},
series = {Lecture Notes in Computer Science (LNCS)},
volume = {10765},
pages = {169--181},
year = 2018,
address = {Newcastle, Australia},
doi = {https://doi.org/10.1007/978-3-319-78825-8_14},
abstract = {Let $S$ be a set of $n$ points in the plane in general position
(no three points from $S$ are collinear).
For a positive integer $k$, a \emph{$k$-hole} in $S$ is a convex polygon
with $k$ vertices from~$S$ and no points of~$S$ in its interior.
For a positive integer $l$, a simple polygon~$P$ is \emph{$l$-convex}
if no straight line intersects the interior of~$P$ in more than $l$ connected components.
A point set $S$ is \emph{$l$-convex} if there exists an $l$-convex polygonization of $S$.
Considering a typical Erd{\H{o}}s--Szekeres-type problem, we show that every 2-convex point
set of size~$n$ contains an $\Omega(\log n)$-hole.
In comparison, it is well known that there exist arbitrarily
large point sets in general position with no 7-hole.
Further, we show that our bound is tight by constructing 2-convex point sets with holes
of size at most $O(\log n)$.},
originalfile = {/geometry/cggg.bib}
}
@article{abhprvv-hi2cps-18,
author = {O.~Aichholzer and M.~Balko and T.~Hackl and A.~Pilz and P.~Ramos and P.~Valtr and B.~Vogtenhuber},
title = {{{Holes in 2-convex point sets}}},
journal = {Computational Geometry: Theory and Applications},
volume = {74},
pages = {38--49},
year = 2018,
doi = {https://doi.org/10.1016/j.comgeo.2018.06.002},
abstract = {Let $S$ be a set of $n$ points in the plane in general position
(no three points from $S$ are collinear).
For a positive integer $k$, a \emph{$k$-hole} in $S$ is a convex polygon
with $k$ vertices from~$S$ and no points of~$S$ in its interior.
For a positive integer $l$, a simple polygon~$P$ is \emph{$l$-convex}
if no straight line intersects the interior of~$P$ in more than $l$ connected components.
A point set $S$ is \emph{$l$-convex} if there exists an $l$-convex polygonization of $S$.
Considering a typical Erd{\H{o}}s--Szekeres-type problem, we show that every 2-convex point
set of size~$n$ contains an $\Omega(\log n)$-hole.
In comparison, it is well known that there exist arbitrarily
large point sets in general position with no 7-hole.
Further, we show that our bound is tight by constructing 2-convex point sets with holes
of size at most $O(\log n)$.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ahs-plsps-15,
author = {O.~Aichholzer and T.~Hackl and M.~Scheucher},
title = {{{Planar L-Shaped Point Set Embedding of Trees}}},
booktitle = {Proc. $32^{st}$ European Workshop on Computational Geometry EuroCG '16},
pages = {51--54},
year = 2016,
address = {Lugano, Switzerland},
category = {3b},
eprint = {},
pdf = {/files/publications/geometry/ahs-plsps-15.pdf},
thackl_label = {51C},
abstract = {In this paper we consider planar L-shaped embeddings of
trees in point sets, that is, planar drawings where
the vertices are mapped to a subset of the given
points and where every edge consists of two
axis-aligned line segments. We investigate the
minimum number $m$, such that any $n$ vertex tree
with maximum degree 4 admits a planar L-shaped
embedding in any point set of size $m$. First we
give an upper bound $O(n^c)$ with $c=log_23
{\approx} 1.585$ for the general case, and thus
answer the question by Di Giacomo et al. whether
a sub- quadratic upper bound exists. Then we
introduce the saturation function for trees and show
that trees with low saturation can be embedded
even more efficiently. In particular, we improve
the upper bound for caterpillars and extend the
class of trees that require only a linear number of
points. In addition, we present some probabilistic
results for either randomly chosen trees or randomly
chosen point sets.},
originalfile = {/geometry/cggg.bib}
}
@article{aafpdfuv-cbitcp-18,
author = {Oswin Aichholzer
and Nieves Atienza
and Jos{\'e} M. D{\'i}az-B{\'a}{\~n}ez
and Ruy Fabila-Monroy
and David Flores-Pe{\~n}aloza
and Pablo P{\'e}rez-Lantero
and Jorge Urrutia
and Birgit Vogtenhuber},
title = {{{Computing Balanced Islands in Two Colored Point Sets in the Plane}}},
journal = {Information Processing Letters},
volume = {135},
pages = {28 -- 32},
year = {2018},
issn = {0020-0190},
doi = {https://doi.org/10.1016/j.ipl.2018.02.008},
url = {http://www.sciencedirect.com/science/article/pii/S0020019018300371},
eprint = {1510.01819},
archiveprefix = {arXiv},
abstract = {Let $S$ be a set of $n$ points in general position in the plane,
$r$ of which are red and $b$ of which are blue.
In this paper we present algorithms to find convex sets containing a balanced number of red and blue points.
We provide an $O(n^4)$ time algorithm that for a given $\alpha \in \left [ 0,\frac{1}{2} \right ]$
finds a convex set containing exactly $\lceil \alpha r\rceil$ red points and exactly $\lceil \alpha b \rceil$
blue points of $S$. If $\lceil \alpha r\rceil+\lceil \alpha b\rceil$ is not much larger than
$\frac{1}{3}n$, we improve the running time to $O(n \log n)$.
We also provide an $O(n^2\log n)$ time algorithm to find a convex set containing exactly
$\left \lceil \frac{r+1}{2}\right \rceil$ red points and exactly $\left \lceil \frac{b+1}{2}\right \rceil$
blue points of $S$, and show that balanced islands with more points do not always exist.},
keywords = {Equipartition, Islands, Convex sets, Computational geometry},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aahpsv-ilbnt-16,
author = {O.~Aichholzer and V.~Alvarez and T.~Hackl and A.~Pilz and B.~Speckmann and B.~Vogtenhuber},
title = {{{An improved lower bound on the number of triangulations}}},
booktitle = {\em Proc. $32^{nd}$ Int. Sympos. Comput. Geom. (SoCG) volume~51 of
Leibniz International Proceedings in Informatics (LIPIcs)},
pages = {7:1--7:16},
year = 2016,
address = {Boston, USA},
category = {3b},
eprint = {},
doi = {10.4230/LIPIcs.SoCG.2016.7},
pdf = {/files/publications/geometry/aahpsv-ilbnt-16.pdf},
thackl_label = {53C},
abstract = {Upper and lower bounds for the number of geometric
graphs of specific types on a given set of points in
the plane have been intensively studied in recent
years. For most classes of geometric graphs it is
now known that point sets in convex position
minimize their number. However, it is still unclear
which point sets minimize the number of geometric
triangulations; the so-called double circles are
conjectured to be the minimizing sets. In this paper
we prove that any set of $n$ points in general
position in the plane has at least $\Omeag(2.631^n)$
geometric triangulations. Our result improves the
previously best general lower bound of $\Omega(2.43^n)$
and also covers the previously best lower bound of
$\Omega(2.63^n)$ for a fixed number of extreme points. We
achieve our bound by showing and combining several
new results, which are of independent interest: 1.
Adding a point on the second convex layer of a given
point set (of $7$ or more points) at least doubles
the number of triangulations. 2. Generalized
configurations of points that minimize the number of
triangulations have at most ${[}n/2{]}$ points on
their convex hull. 3. We provide tight lower
bounds for the number of triangulations of point
sets with up to 15 points. These bounds further
support the double circle conjecture.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{abhkpsvv-slbnh-17,
author = {O.~Aichholzer and M.~Balko and T.~Hackl and J.~Kyn\v{c}l and
I.~Parada and M.~Scheucher and P.~Valtr and B.~Vogtenhuber},
title = {{{A superlinear lower bound on the number of 5-holes}}},
booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)},
pages = {8:1--8:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
volume = {77},
editor = {Boris Aronov and Matthew J. Katz},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
year = 2017,
address = {Brisbane, Australia},
category = {3b},
eprint = {1703.05253},
archiveprefix = {arXiv},
doi = {10.4230/LIPIcs.SoCG.2017.8},
pdf = {/files/publications/geometry/abhkpsvv-slbnh-17.pdf},
abstract = {Let $P$ be a finite set of points in the plane in
\emph{general position}, that is, no three points of
$P$ are on a common line. We say that a set $H$ of
five points from $P$ is a \emph{$5$-hole in~$P$} if
$H$ is the vertex set of a convex $5$-gon containing
no other points of~$P$. For a positive integer $n$,
let $h_5(n)$ be the minimum number of 5-holes among
all sets of $n$ points in the plane in general
position. Despite many efforts in the last 30
years, the best known asymptotic lower and upper
bounds for $h_5(n)$ have been of order $\Omega(n)$
and~$O(n^2)$, respectively. We show that $h_5(n) =
\Omega(n\log^{4/5}{n})$, obtaining the first
superlinear lower bound on $h_5(n)$. The following
structural result, which might be of independent
interest, is a crucial step in the proof of this
lower bound. If a finite set $P$ of points in the
plane in general position is partitioned by a line
$\ell$ into two subsets, each of size at least 5 and
not in convex position, then $\ell$ intersects the
convex hull of some 5-hole in~$P$. The proof of
this result is computer-assisted.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{abhkpsvv-slbnh-17a,
author = {O.~Aichholzer and M.~Balko and T.~Hackl and J.~Kyn\v{c}l and
I.~Parada and M.~Scheucher and P.~Valtr and B.~Vogtenhuber},
title = {{{A superlinear lower bound on the number of 5-holes}}},
booktitle = {Proc. $33^{rd}$ European Workshop on Computational Geometry EuroCG '17},
pages = {69--73},
year = 2017,
address = {Malm\"o, Sweden},
category = {3b},
eprint = {},
pdf = {/files/publications/geometry/abhkpsvv-slbnh-17a.pdf},
abstract = {Let $P$ be a finite set of points in the plane in
\emph{general position}, that is, no three points of
$P$ are on a common line. We say that a set $H$ of
five points from $P$ is a \emph{$5$-hole in~$P$} if
$H$ is the vertex set of a convex $5$-gon containing
no other points of~$P$. For a positive integer $n$,
let $h_5(n)$ be the minimum number of 5-holes among
all sets of $n$ points in the plane in general
position. Despite many efforts in the last 30
years, the best known asymptotic lower and upper
bounds for $h_5(n)$ have been of order $\Omega(n)$
and~$O(n^2)$, respectively. We show that $h_5(n) =
\Omega(n\log^{4/5}{n})$, obtaining the first
superlinear lower bound on $h_5(n)$. The following
structural result, which might be of independent
interest, is a crucial step in the proof of this
lower bound. If a finite set $P$ of points in the
plane in general position is partitioned by a line
$\ell$ into two subsets, each of size at least 5 and
not in convex position, then $\ell$ intersects the
convex hull of some 5-hole in~$P$. The proof of
this result is computer-assisted.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aabv-pcmgt-17,
author = {O.~Aichholzer and L.~Andritsch and K.~Baur and B.~Vogtenhuber},
title = {{{Perfect $k$-colored matchings and $k+2$-gonal tilings}}},
booktitle = {Proc. $33^{rd}$ European Workshop on Computational Geometry EuroCG '17},
pages = {81--84},
year = 2017,
address = {Malm\"o, Sweden},
category = {3b},
eprint = {1710.06757},
archiveprefix = {arXiv},
pdf = {/files/publications/geometry/aabv-pcmgt-17.pdf},
abstract = { We derive a simple bijection between geometric plane perfect
matchings on $2n$ points in convex position and triangulations on
$n+2$ points in convex position. We then extend this bijection to
monochromatic plane perfect matchings on periodically $k$-colored
vertices and $(k+2)$-gonal tilings of convex point sets. These
structures are related to Temperley-Lieb algebras and our bijections
provide explicit one-to-one relations between matchings and tilings.
Moreover, for a given element of one class, the corresponding
element of the other class can be computed in linear time.},
originalfile = {/geometry/cggg.bib}
}
@article{ahkklpsw-ppstp-17,
author = {O.~Aichholzer and T.~Hackl and M.~Korman and M.~van~Kreveld and
M.~L\"offler and A.~Pilz and B.~Speckmann and E.~Welzl},
title = {{{Packing Plane Spanning Trees and Paths in Complete Geometric Graphs}}},
journal = {Information Processing Letters (IPL)},
volume = {124},
pages = {35--41},
year = 2017,
category = {3a},
arxiv = {1707.05440},
doi = {http://dx.doi.org/10.1016/j.ipl.2017.04.006},
pdf = {/files/publications/geometry/ahkklpsw-ppstp-17.pdf},
abstract = {We consider the following question: How many
edge-disjoint plane spanning trees are contained in a complete
geometric graph $GK_n$ on any set $S$ of $n$ points in general
position in the plane?},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aafmm-bdk-17,
author = {Bernardo M. {\'A}brego and Oswin Aichholzer and Silvia Fern{\'a}ndez-Merchant and
Dan McQuillan and Bojan Mohar and Petra Mutzel and Pedro Ramos and R. Bruce Richter and Birgit Vogtenhuber},
title = {{{Bishellable drawings of $K_n$}}},
booktitle = {Proc. XVII Encuentros de Geometr\'{\i}a Computacional},
category = {3b},
pages = {17--20},
pdf = {/files/publications/geometry/aafmm-bdk-17.pdf},
year = 2017,
address = {Alicante, Spain},
eprint = {1510.00549},
abstract = {In this work, we generalize the concept of
$s$-shellability to bishellability, where the former
implies the latter in the sense that every
$s$-shellable drawing is, for any $b \leq s-2$, also
$b$-bishellable. Our main result is that
$(\lfloor \frac{n}{2} \rfloor\!-\!2)$-bishellability
also guarantees, with a simpler proof than for
$s$-shellability, that a drawing has at least
$H(n)$ crossings. We exhibit a drawing of $K_{11}$
that has $H(11)$ crossings, is 3-bishellable, and is
not $s$-shellable for any $s\geq5$. This shows that
we have properly extended the class of drawings for
which the Harary-Hill Conjecture is proved.},
originalfile = {/geometry/cggg.bib}
}
@article{aafmm-bdk-18,
author = {Bernardo M. {\'A}brego and Oswin Aichholzer and Silvia Fern{\'a}ndez-Merchant and
Dan McQuillan and Bojan Mohar and Petra Mutzel and Pedro Ramos and R. Bruce Richter and Birgit Vogtenhuber},
title = {{{Bishellable drawings of {$K_n$}}}},
noshortjournal = {SIAM J. Discrete Math.},
journal = {SIAM Journal on Discrete Mathematics},
volume = {32},
number = {4},
pages = {2482--2492},
year = {2018},
doi = {10.1137/17M1147974},
eprint = {1510.00549},
archiveprefix = {arXiv},
abstract = {{The Harary--Hill conjecture, still open after more than 50 years,
asserts that the crossing number of the complete graph $K_n$ is
$H(n) := \frac 1 4 \left\lfloor\frac{\mathstrut n}{\mathstrut 2}\right\rfloor
\left\lfloor\frac{\mathstrut n-1}{\mathstrut 2}\right\rfloor
\left\lfloor\frac{\mathstrut n-2}{\mathstrut 2}\right\rfloor
\left\lfloor\frac{\mathstrut n-3}{\mathstrut 2}\right\rfloor\,.$
\'Abrego et al. [B.~M. {\'{A}}brego, O. Aichholzer, S. Fern{\'{a}}ndez{-}Merchant,
P. Ramos, and G. Salazar. Shellable drawings and the cylindrical crossing number of
$K_n$. {\em Disc. {\&} Comput. Geom.}, 52(4):743--753, 2014.]
introduced the notion of shellability of a drawing $D$ of $K_n$.
They proved that if $D$ is $s$-shellable for some $s\geq\lfloor\frac{n}{2}\rfloor$,
then $D$ has at least $H(n)$ crossings.
This is the first combinatorial condition on a drawing
that guarantees at least $H(n)$ crossings.
In this work, we generalize the concept of $s$-shellability to bishellability,
where the former implies the latter in the sense that every $s$-shellable drawing is,
for any $b \leq s-2$, also \mbox{$b$-bishellable}.
Our main result is that $(\lfloor \frac{n}{2} \rfloor\!-\!2)$-bishellability of a drawing
$D$ of $K_n$ also guarantees, with a simpler proof than for \mbox{$s$-shellability},
that $D$ has at least $H(n)$ crossings.
We exhibit a drawing of $K_{11}$ that has $H(11)$ crossings, is 3-bishellable, and is not
$s$-shellable for any $s\geq5$.This shows that we have properly extended the class of
drawings for which the Harary--Hill Conjecture is proved. Moreover, we provide an infinite
family of drawings of $K_n$ that are $(\lfloor \frac{n}{2} \rfloor\!-\!2)$-bishellable,
but not $s$-shellable for any $s\geq\lfloor\frac{n}{2}\rfloor$.}},
originalfile = {/geometry/cggg.bib}
}
@article{affmpps-mmvqt-17,
title = {{{Minimization and Maximization Versions of the Quadratic Traveling Salesman Problem}}},
author = {O.~Aichholzer and A.~Fischer and F.~Fischer J.F.~Meier and U.~Pferschy and A.~Pilz and R.~Stanek},
journal = {OPTIMIZATION},
doi = {http://dx.doi.org/10.1080/02331934.2016.1276905},
volume = {66},
number = {4},
pages = {521--546},
year = {2017},
publisher = {TAYLOR \& FRANCIS LTD 2-4 PARK SQUARE, MILTON PARK, ABINGDON OR14 4RN, OXON, ENGLAND},
pdf = {/files/publications/geometry/affmpps-mmvqt-17.pdf},
abstract = {The traveling salesman problem (TSP) asks for a shortest
tour through all vertices of a graph with respect to
the weights of the edges. The symmetric quadratic
traveling salesman problem (SQTSP) associates a
weight with every three vertices traversed in
succession. If these weights correspond to the
turning angles of the tour, we speak of the
angular-metric traveling salesman problem (Angle
TSP). In this paper, we first consider the SQTSP
from a computational point of view. In particular,
we apply a rather basic algorithmic idea and perform
the separation of the classical subtour elimination
constraints on integral solutions
only. Surprisingly, it turns out that this approach
is faster than the standard fractional separation
procedure known from the literature. We also test
the combination with strengthened subtour
elimination constraints for both variants, but these
turn out to slow down the computation. Secondly, we
provide a completely different, mathematically
interesting MILP linearization for the Angle TSP
that needs only a linear number of additional
variables while the standard linearization requires
a cubic one. For medium sized instances of a variant
of the Angle TSP this formulation yields reduced
running times. However, for larger instances or pure
Angle TSP instances the new formulation takes more
time to solve than the known standard model.
Finally, we introduce MaxSQTSP, the maximization
version of the quadratic traveling salesman
problem. Here it turns out that using some of the
stronger subtour elimination constraints helps. For
the special case of the MaxAngle TSP we can observe
an interesting geometric property if the number of
vertices is odd. We show that the sum of inner
turning angles in an optimal solution always equals
$\pi$. This implies that the problem can be solved
by the standard ILP model without producing any
integral subtours. Moreover, we give a simple
constructive polynomial time algorithm to find such
an optimal solution. If the number of vertices is
even the optimal value lies between 0 and $2\pi$ and
these two bounds are tight, which can be shown by an
analytic solution for a regular $n$-gon.},
originalfile = {/geometry/cggg.bib}
}
@article{abhpv-ltdbm-17,
author = {Oswin Aichholzer and Luis Barba and Thomas Hackl and Alexander Pilz and Birgit Vogtenhuber},
title = {{{Linear Transformation Distance for Bichromatic
Matchings}}},
journal = {Computational Geometry: Theory and Applications},
year = 2018,
volume = {68},
pages = {77--88},
category = {3a},
doi = {http://dx.doi.org/10.1016/j.comgeo.2017.05.003},
arxiv = {1312.0884v1},
note = {Special Issue in Memory of Ferran Hurtado},
issn = {0925-7721},
url = {http://www.sciencedirect.com/science/article/pii/S0925772117300366},
pdf = {/files/publications/geometry/abhpv-ltdbm-17.pdf},
abstract = {Let $P=B\cup R$ be a set of $2n$ points in general
position, where $B$ is a set of $n$ blue points and
$R$ a set of $n$ red points. A \emph{$BR$-matching}
is a plane geometric perfect matching on $P$ such
that each edge has one red endpoint and one blue
endpoint. Two $BR$-matchings are compatible if their
union is also plane.\\ The \emph{transformation
graph of $BR$-matchings} contains one node for each
$BR$-matching and an edge joining two such nodes if
and only if the corresponding two $BR$-matchings are
compatible. In SoCG 2013 it has been shown by
Aloupis, Barba, Langerman, and Souvaine that this
transformation graph is always connected, but its
diameter remained an open question. In this paper we
provide an alternative proof for the connectivity of
the transformation graph and prove an upper bound of
$2n$ for its diameter, which is asymptotically
tight.},
originalfile = {/geometry/cggg.bib}
}
@article{affhhuv-mimp-17,
author = {Oswin Aichholzer and Ruy Fabila-Monroy and David Flores-Pe{\~n}aloza and Thomas Hackl and Jorge Urrutia and Birgit Vogtenhuber},
title = {{{Modem Illumination of Monotone Polygons}}},
journal = {Computational Geometry: Theory and Applications},
year = 2018,
volume = {68},
number = {},
pages = {101--118},
category = {3a},
doi = {https://doi.org/10.1016/j.comgeo.2017.05.010},
arxiv = {1503.05062},
note = {Special Issue in Memory of Ferran Hurtado},
issn = {0925-7721},
url = {http://www.sciencedirect.com/science/article/pii/S0925772117300433},
abstract = {We study a generalization of the classical problem of
illumination of polygons. Instead of modeling a light
source we model a wireless device whose radio signal can
penetrate a given number $k$ of walls. We call these
objects $k$-modems and study the minimum number of
$k$-modems necessary to illuminate monotone and monotone
orthogonal polygons. We show that every monotone polygon on
$n$ vertices can be illuminated with $\left\lceil
\frac{n}{2k} \right\rceil$ $k$-modems and exhibit examples
of monotone polygons requiring $\left\lceil \frac{n}{2k+2}
\right\rceil$ $k$-modems. For monotone orthogonal polygons,
we show that every such polygon on $n$ vertices can be
illuminated with $\left\lceil \frac{n}{2k+4} \right\rceil$
$k$-modems and give examples which require $\left\lceil
\frac{n}{2k+4} \right\rceil$ $k$-modems for $k$ even and
$\left\lceil \frac{n}{2k+6} \right\rceil$ for $k$ odd.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{abhkmppsvvw-mggrot-18,
author = {Oswin Aichholzer and
Martin Balko and
Michael Hoffmann and
Jan Kyn\v{c}l and
Wolfgang Mulzer and
Irene Parada and
Alexander Pilz and
Manfred Scheucher and
Pavel Valtr and
Birgit Vogtenhuber and
Emo Welzl},
title = {{{Minimal Geometric Graph Representations of Order Types}}},
booktitle = {Proc. {$34^{th}$} European Workshop on Computational Geometry EuroCG '18},
year = 2018,
pages = {21:1--21:6},
address = {Berlin, Germany},
abstract = {We consider the problem of characterizing small geometric graphs
whose structure uniquely determines the order type of its vertex set.
We describe a set of edges that prevent the order type from changing
by continuous movement and identify properties of the resulting graphs.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aktv-npmt-18,
author = {Oswin Aichholzer and
Michael Kerber and
Istv{\'a}n Talata and
Birgit Vogtenhuber},
title = {{{A Note on Planar Monohedral Tilings}}},
booktitle = {Proc. {$34^{th}$} European Workshop on Computational Geometry EuroCG '18},
year = 2018,
pages = {31:1--31:6},
address = {Berlin, Germany},
abstract = {{A planar \emph{monohedral tiling} is a decomposition of $\mathbb{R}^2$
into congruent \emph{tiles}.
We say that such a tiling has the \emph{flag property} if for each triple
of tiles that intersect pairwise, the three tiles intersect in a common point.
We show that for convex tiles, there exist only three classes of tilings
that are not flag, and they all consist of triangular tiles; in particular,
each convex tiling using polygons with $n\geq 4$ vertices is flag.
We also show that an analogous statement for the case of non-convex tiles
is not true by presenting a family of counterexamples.}},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{amsv-mcsig-18,
author = {Oswin Aichholzer and
Wolfgang Mulzer and
Partick Schnider and
Birgit Vogtenhuber},
title = {{{NP-Completeness of Max-Cut for Segment Intersection Graphs}}},
booktitle = {Proc. {$34^{th}$} European Workshop on Computational Geometry EuroCG '18},
year = 2018,
pages = {32:1-32:6},
address = {Berlin, Germany},
abstract = {{We consider the problem of finding a \emph{maximum cut} in a graph $G = (V, E)$,
that is, a partition $ V_1 \dot\cup V_2$ of $V$ such that the number of edges between $V_1$ and $V_2$ is maximum.
It is well known that the decision problem whether $G$ has a cut of at least a given size is in general NP-complete.
We show that this problem remains hard when restricting the input to \emph{segment intersection graphs}.
These are graphs whose vertices can be drawn as straight-line segments,
where two vertices share an edge if and only if the corresponding segments intersect.
We obtain our result by a reduction from a variant of \textsc{Planar Max-2-SAT}
that we introduce and also show to be NP-complete.}},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aepv-assdcg-17,
author = {Oswin Aichholzer and
Florian Ebenf\"{u}hrer and
Irene Parada and
Alexander Pilz and
Birgit Vogtenhuber},
title = {{{On semi-simple drawings of the complete graph}}},
booktitle = {Proc. XVII Encuentros de Geometr\'{\i}a Computacional},
pages = {25--28},
year = 2017,
address = {Alicante, Spain},
pdf = {/files/publications/geometry/aepv-assdcg-17.pdf},
abstract = {In this work we study rotation systems and semi-simple drawings of $K_n$. A simple drawing of a graph is a drawing in which every pair of edges intersects in at most one point. In a semi-simple drawing, edge pairs might intersect in multiple points, but incident edges only intersect in their common endpoint. A rotation system is called (semi-)realizable if it can be realized with a (semi-)simple drawing. It is known that a rotation system is realizable if and only if all its 5-tuples are realizable. For the problem of characterizing semi-realizability, we present a rotation system with six vertices that is not semi-realizable, although all its 5-tuples are semi-realizable. Moreover, by an exhaustive computer search, we show that also for seven vertices there exist minimal not semi-realizable rotation systems (that is, rotation systems in which all proper sub-rotation systems are semi-realizable). This indicates that checking semi-realizability is harder than checking realizability. Finally we show that for semi-simple drawings, generalizations of Conway's Thrackle Conjecture and the conjecture on the existence of plane Hamiltonian cycles do not hold.},
originalfile = {/geometry/cggg.bib}
}
@article{aabv-pcmgt-18,
author = {Oswin Aichholzer and Lukas Andritsch and Karin Baur and Birgit Vogtenhuber},
title = {{{Perfect $k$-Colored Matchings and $(k+2)$-Gonal Tilings}}},
journal = {Graphs and Combinatorics},
issn = {0911-0119},
publisher = {Springer Japan},
pages = {1333--1346},
volume = {34},
number = {6},
doi = {https://doi.org/10.1007/s00373-018-1967-8},
url = {http://link.springer.com/article/10.1007/s00373-018-1967-8},
year = 2018,
eprint = {1710.06757},
archiveprefix = {arXiv},
abstract = {We derive a simple bijection between geometric plane perfect matchings on $2n$ points in convex position and triangulations on $n+2$ points in convex position. We then extend this bijection to monochromatic plane perfect matchings on periodically $k$-colored vertices and $(k+2)$-gonal tilings of convex point sets. These structures are related to a generalization of Temperley–Lieb algebras and our bijections provide explicit one-to-one relations between matchings and tilings. Moreover, for a given element of one class, the corresponding element of the other class can be computed in linear time.},
originalfile = {/geometry/cggg.bib}
}
@article{akmpw-oarps-17,
author = {O. Aichholzer and V. Kusters and W. Mulzer and A. Pilz and M. Wettstein},
title = {{{An optimal algorithm for reconstructing point set order types from radial orderings}}},
journal = {International Journal of Computational Geometry \& Applications},
year = 2017,
volume = {27},
number = {1--2},
pages = {57--83},
doi = {10.1142/S0218195917600044},
url = {http://www.scopus.com/inward/record.url?scp=85029349312&partnerID=8YFLogxK},
eprint = {1507.08080},
abstract = {Let $P$ be a set of $n$ labeled points in the plane.
The \emph{radial system} of~$P$ describes, for each
$p\in P$, the order in which a ray that rotates
around $p$ encounters the points in $P \setminus \{p\}$.
This notion is related to the \emph{order type}
of~$P$, which describes the orientation (clockwise
or counterclockwise) of every ordered triple in~$P$.
Given only the order type, the radial
system is uniquely determined and can
easily be obtained. The converse, however,
is not true. Indeed, let $R$ be the radial system
of $P$, and let $T(R)$ be the set of all order
types with radial system $R$
(we define $T(R) = \emptyset$ for the
case that $R$ is not a valid radial system).
Aichholzer \etal~(\emph{Reconstructing
Point Set Order Types from Radial Orderings}, in
Proc.~ISAAC 2014) show that
$T(R)$ may contain up to $n-1$ order types.
They also provide polynomial-time algorithms to
compute $T(R)$ when only $R$ is given.
We describe a new algorithm for
finding $T(R)$. The algorithm constructs the convex
hulls of all possible point sets with the radial
system $R$. After that, orientation queries on point triples
can be answered in constant time. A representation of this set of convex
hulls can be found in $O(n)$ queries to the radial system,
using $O(n)$ additional processing time. This is optimal.
Our results also generalize
to \emph{abstract order types}.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ap-tcep-19,
author = {Oswin Aichholzer and Daniel Perz},
title = {{{Triangles in the colored Euclidean plane}}},
booktitle = {Proc. {$35^{th}$} European Workshop on Computational Geometry EuroCG '19},
year = 2019,
pages = {10:1-10:7},
address = {Utrecht, The Netherlands},
pdf = {/files/publications/geometry/ap-tcep-19.pdf},
url = {http://www.eurocg2019.uu.nl/papers/10.pdf},
abstract = {{We study a variation of the well known Hadwiger-Nelson problem on the chromatic number
of the Euclidean plane. An embedding of a given triangle $T$ into the colored plane is called
monochromatic, if the three corners of the triangle get the same color. We provide a classification
of triangles according to the number of colors needed to color the plane so that the triangle can
not be embedded monochromatically. For example, we show that for near-equilateral triangles
three colors are enough and that for almost all triangles six colors are sufficient.}},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{adhopvv-estg-19,
author = {Oswin Aichholzer and Jos{\'e} Miguel D\'{\i}az-B{\'a}{\~n}ez and Thomas Hackl and David Orden and Alexander Pilz and Inmaculada Ventura and Birgit Vogtenhuber},
title = {{{Erd{\H{o}}s-Szekeres-Type Games}}},
booktitle = {Proc. {$35^{th}$} European Workshop on Computational Geometry EuroCG '19},
year = 2019,
pages = {23:1-23:7},
address = {Utrecht, The Netherlands},
pdf = {/files/publications/geometry/adhopvv-estg-19.pdf},
url = {http://www.eurocg2019.uu.nl/papers/23.pdf},
abstract = {{We consider several combinatorial games, inspired by the Erd{\H{o}}s-Szekeres theorem that states the
existence of a convex $k$-gon in every sufficiently large point set. Two players take turns to place
points in the Euclidean plane and the game is over as soon as the first $k$-gon appears. In the
Maker-Maker setting the player who placed the last point wins, while in the Avoider-Avoider
version this player loses. Combined versions like Maker-Breaker are also possible. Moreover,
variants can be obtained by considering that (1) the points to be placed are either uncolored or
bichromatic, (2) both players have their own color or can play with both colors, (3) the
$k$-gon must be empty of other points, or (4) the $k$-gon has to be convex.}},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{affhpvz-ccn-19,
author = {Oswin Aichholzer and Ruy Fabila Monroy and Adrian Fuchs and Carlos Hidalgo Toscano and Irene Parada and Birgit Vogtenhuber and Francisco Zaragoza},
title = {{{On the 2-Colored Crossing Number}}},
booktitle = {Proc. {$35^{th}$} European Workshop on Computational Geometry EuroCG '19},
year = 2019,
pages = {56:1-56:7},
address = {Utrecht, The Netherlands},
pdf = {/files/publications/geometry/affhpvz-ccn-19.pdf},
url = {http://www.eurocg2019.uu.nl/papers/56.pdf},
abstract = {{Let $D$ be a straight-line drawing of a graph where every edge is colored with one of two possible
colors. The rectilinear 2-colored crossing number of $D$ is the minimum number of crossings
between edges of the same color, taken over all possible colorings of $D$. We show lower and upper
bounds on the rectilinear 2-colored crossing number for the complete graph $K_n$. Moreover, for
fixed drawings of $K_n$ we give bounds on the relation between its rectilinear 2-colored crossing
number and its rectilinear crossing number.}},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{apsvw-sssdk-19,
author = {Oswin Aichholzer and Irene Parada and Manfred Scheucher and Birgit Vogtenhuber and Alexandra Weinberger},
title = {{{Shooting Stars in Simple Drawings of $K_{m,n}$}}},
booktitle = {Proc. {$35^{th}$} European Workshop on Computational Geometry EuroCG '19},
year = 2019,
pages = {59:1-59:6},
address = {Utrecht, The Netherlands},
pdf = {/files/publications/geometry/apsvw-sssdk-19.pdf},
url = {http://www.eurocg2019.uu.nl/papers/59.pdf},
eprint = {2209.01190},
archiveprefix = {arXiv},
abstract = {{In this work we study the existence of plane spanning trees in simple drawings of the complete
bipartite graph $K_{m,n}$. We show that every simple drawing of $K_{2,n}$ and $K_{3,n}$, $n \geq 1$, as well as
every outer drawing of $K_{m,n}$ for any $m,n \geq 1$, contains plane spanning trees. Moreover, for all
these cases we show the existence of special plane spanning trees, which we call shooting stars.
Shooting stars are spanning trees that contain the star of a vertex, i.e., all its incident edges.}},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aappttuv-hmpcb-19,
author = {Oswin Aichholzer and Carlos Alegr\'{i}a Galicia and Irene Parada and Alexander Pilz and Javier Tejel and Csaba D. T\'{o}th and Jorge Urrutia and Birgit Vogtenhuber},
title = {{{Hamiltonian meander paths and cycles on bichromatic point sets}}},
booktitle = {Proc. XVIII Encuentros de Geometr\'{\i}a Computacional},
pages = {35--38},
year = 2019,
address = {Girona, Spain},
pdf = {/files/publications/geometry/aappttuv-hmpcb-19.pdf},
url = {http://imae.udg.edu/egc2019/doc/BookAbstractsEGC2019.pdf},
abstract = {We show that any set of $n$ blue and $n$ red points on a
line admits a plane meander path, that is, a crossing-free
panning path that passes across the line on red
and blue points in alternation. For meander cycles,
we derive tight bounds on the minimum number of
necessary crossings which depend on the coloring of
the points. Finally, we provide some relations for the
number of plane meander paths.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aappttuv-hmpcb2-19,
author = {Oswin Aichholzer and Carlos Alegr\'{i}a Galicia and Irene Parada and Alexander Pilz and Javier Tejel and Csaba D. T\'{o}th and Jorge Urrutia and Birgit Vogtenhuber},
title = {{{Hamiltonian meander paths and cycles on bichromatic point sets}}},
booktitle = {EasyChair Preprint no. 3130},
year = 2019,
url = {https://easychair.org/publications/preprint/N7TX},
abstract = {We show that any set of $n$ blue and $n$ red points on a
line admits a plane meander path, that is, a crossing-free
panning path that passes across the line on red
and blue points in alternation. For meander cycles,
we derive tight bounds on the minimum number of
necessary crossings which depend on the coloring of
the points. Finally, we provide some relations for the
number of plane meander paths.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{afo-tv-19,
author = {Oswin Aichholzer and Ruy Fabila-Monroy and Julia Obmann},
title = {{{On the Triangle Vector}}},
booktitle = {Proc. XVIII Encuentros de Geometr\'{\i}a Computacional},
pages = {55--58},
year = 2019,
address = {Girona, Spain},
pdf = {/files/publications/geometry/afo-tv-19.pdf},
url = {http://imae.udg.edu/egc2019/doc/BookAbstractsEGC2019.pdf},
abstract = {Let $S$ be a set of $n$ points in the plane in general position.
In this note we study the so-called triangle vector $\tau$ of~$S$.
For each cardinality $i$, $0 \leq i \leq n-3$, $\tau(i)$ is the number of triangles spanned by points of $S$ which contain exactly $i$ points of $S$ in their interior.
We show relations of this vector to other combinatorial structures and derive tight upper bounds for several entries of $\tau$, including $\tau(n-6)$ to $\tau(n-3)$.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{affhpvz-ccn-19b,
author = {Oswin Aichholzer and Ruy Fabila-Monroy and Adrian Fuchs and Carlos Hidalgo-Toscano and Irene Parada and Birgit Vogtenhuber and Francisco Zaragoza},
title = {{{On the 2-colored crossing number}}},
booktitle = {Graph Drawing and Network Visualization. GD 2019},
nonote = {27th International Symposium on Graph Drawing and Network Visualization},
series = {Lecture Notes in Computer Science (LNCS)},
volume = {11904},
pages = {87--100},
address = {Prague, Czechia},
year = 2019,
eprint = {1908.06461},
archiveprefix = {arXiv},
doi = {https://doi.org/10.1007/978-3-030-35802-0_7},
abstract = {Let $D$ be a straight-line drawing of a graph.
The rectilinear 2-colored crossing number of $D$ is the minimum number of crossings between edges of the same color,
taken over all possible 2-colorings of the edges of $D$.
First, we show lower and upper bounds on the rectilinear 2-colored crossing number for the complete graph $K_n$.
To obtain this result, we prove that asymptotic bounds can be derived from optimal and near-optimal instances with few vertices.
We obtain such instances using a combination of heuristics and integer programming.
Second, for any fixed drawing of $K_n$,
we improve the bound on the ratio between its rectilinear 2-colored crossing number and its rectilinear crossing number.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{abhkmppsvvw-mrotg-19,
author = {Oswin Aichholzer and Martin Balko and Michael Hoffmann and Jan Kyn\v{c}l and Wolfgang Mulzer and Irene Parada and Alexander Pilz and Manfred Scheucher and Pavel Valtr and Birgit Vogtenhuber and Emo Welzl},
title = {{{Minimal representations of order types by geometric graphs}}},
booktitle = {Graph Drawing and Network Visualization. GD 2019},
nonote = {27th International Symposium on Graph Drawing and Network Visualization},
series = {Lecture Notes in Computer Science (LNCS)},
volume = {11904},
pages = {101--113},
address = {Prague, Czechia},
year = 2019,
eprint = {1908.05124},
doi = {https://doi.org/10.1007/978-3-030-35802-0_8},
abstract = {In order to have a compact visualization of the order type of
a given point set $S$,
we are interested in geometric graphs on $S$ with few edges that unequivocally display %
the order type of $S$.
We introduce the concept of \emph{exit edges},
which prevent the order type from changing under continuous motion of vertices.
Exit edges have a natural dual characterization,
which allows us to efficiently compute them and to bound their number.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{akopprv-gwlta-19b,
author = {Oswin Aichholzer and Matias Korman and Yoshio Okamoto and Irene Parada and Daniel Perz and Andr\'e van Renssen and Birgit Vogtenhuber},
title = {{{Graphs with large total angular resolution}}},
booktitle = {Graph Drawing and Network Visualization. GD 2019},
nonote = {27th International Symposium on Graph Drawing and Network Visualization},
series = {Lecture Notes in Computer Science (LNCS)},
volume = {11904},
pages = {193--199},
address = {Prague, Czechia},
year = 2019,
eprint = {1908.06504},
archiveprefix = {arXiv},
doi = {https://doi.org/10.1007/978-3-030-35802-0_15},
isbn = {978-3-030-35802-0},
abstract = {The total angular resolution of a straight-line drawing is the minimum angle between two edges of the drawing.
It combines two properties contributing to the readability of a drawing:
the angular resolution, that is the minimum angle between incident edges,
and the crossing resolution, that is the minimum angle between crossing edges.
We consider the total angular resolution of a graph,
which is the maximum total angular resolution of a straight-line drawing of this graph.
We prove that, up to a finite number of well specified exceptions of constant size,
the number of edges of a graph with $n$ vertices and a total angular resolution greater than $60^{\circ}$ is bounded by $2n-6$.
This bound is tight.
In addition, we show that deciding whether a graph has total angular resolution at least $60^{\circ}$ is \NP-hard.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{akksv-evrmt-19,
author = {Oswin Aichholzer and Linda Kleist and Boris Klemz and Felix Schr\"oder and Birgit Vogtenhuber},
title = {{{On the Edge-Vertex Ratio of Maximal Thrackles}}},
booktitle = {Graph Drawing and Network Visualization. GD 2019},
nonote = {27th International Symposium on Graph Drawing and Network Visualization},
series = {Lecture Notes in Computer Science (LNCS)},
volume = {11904},
pages = {482--495 },
address = {Prague, Czechia},
year = 2019,
eprint = {1908.08857},
doi = {https://doi.org/10.1007/978-3-030-35802-0_37},
abstract = {A drawing of a graph in the plane is a \emph{thrackle} if every pair of edges intersects exactly once,
either at a common vertex or at a proper crossing. Conway's conjecture states that a thrackle has at most as many edges as vertices.
In this paper, we investigate the edge-vertex ratio of \emph{maximal thrackles}, that is, thrackles in which no edge between already
existing vertices can be inserted such that the resulting drawing remains a thrackle. For maximal geometric and topological thrackles,
we show that the edge-vertex ratio can be arbitrarily small. When forbidding isolated vertices, the edge-vertex ratio of maximal
geometric thrackles can be arbitrarily close to the natural lower bound of $\frac{1}{2}$. For maximal topological thrackles without
isolated vertices, we present an infinite family with an edge-vertex ratio arbitrary close to~$\frac{4}{5}$.},
originalfile = {/geometry/cggg.bib}
}
@article{ahkprrrv-ppsgs-19,
author = {Oswin Aichholzer and Thomas Hackl and Matias Korman and Alexander Pilz and Andr{\'e} van Renssen and Marcel Roeloffzen and G\"unter Rote and Birgit Vogtenhuber},
title = {{{Packing plane spanning graphs with short edges in complete geometric graphs}}},
journal = {Computational Geometry},
volume = {782},
pages = {1--15},
year = {2019},
issn = {0925-7721},
doi = {https://doi.org/10.1016/j.comgeo.2019.04.001},
url = {http://www.sciencedirect.com/science/article/pii/S0925772119300495},
pdf = {/files/publications/geometry/ahkprrrv-ppsgs-19.pdf},
abstract = {Given a set of points in the plane, we want to establish a connected spanning graph between these points, called connection network, that consists of several disjoint layers. Motivated by sensor networks, our goal is that each layer is connected, spanning, and plane. No edge in this connection network is too long in comparison to the length needed to obtain a spanning tree. We consider two different approaches. First we show an almost optimal centralized approach to extract two layers. Then we consider a distributed model in which each point can compute its adjacencies using only information about vertices at most a predefined distance away. We show a constant factor approximation with respect to the length of the longest edge in the graphs. In both cases the obtained layers are plane.},
originalfile = {/geometry/cggg.bib}
}
@article{abhkpsvv-slbnh-19,
author = {O.~Aichholzer and M.~Balko and T.~Hackl and J.~Kyn\v{c}l and
I.~Parada and M.~Scheucher and P.~Valtr and B.~Vogtenhuber},
title = {{{A superlinear lower bound on the number of 5-holes}}},
journal = {Journal of Combinatorial Theory A},
year = 2019,
pages = {1--31},
note = {online},
eprint = {1703.05253},
doi = {10.1016/j.jcta.2020.105236},
url = {https://doi.org/10.1016/j.jcta.2020.105236},
pdf = {/files/publications/geometry/abhkpsvv-slbnh-17.pdf},
abstract = {Let $P$ be a finite set of points in the plane in
\emph{general position}, that is, no three points of
$P$ are on a common line. We say that a set $H$ of
five points from $P$ is a \emph{$5$-hole in~$P$} if
$H$ is the vertex set of a convex $5$-gon containing
no other points of~$P$. For a positive integer $n$,
let $h_5(n)$ be the minimum number of 5-holes among
all sets of $n$ points in the plane in general
position. Despite many efforts in the last 30
years, the best known asymptotic lower and upper
bounds for $h_5(n)$ have been of order $\Omega(n)$
and~$O(n^2)$, respectively. We show that $h_5(n) =
\Omega(n\log^{4/5}{n})$, obtaining the first
superlinear lower bound on $h_5(n)$. The following
structural result, which might be of independent
interest, is a crucial step in the proof of this
lower bound. If a finite set $P$ of points in the
plane in general position is partitioned by a line
$\ell$ into two subsets, each of size at least 5 and
not in convex position, then $\ell$ intersects the
convex hull of some 5-hole in~$P$. The proof of
this result is computer-assisted.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{achkmsv-fdbgo-19,
author = {Oswin Aichholzer and Jean Cardinal and Tony Huynh and Kolja Knauer and Torsten M{\"u}tze and Raphael Steiner and Birgit Vogtenhuber},
title = {{{Flip distances between graph orientations}}},
booktitle = {45th International Workshop on Graph-Theoretic Concepts in Computer Science},
pages = {120--134},
address = {Vall de Nuria, Spain},
series = {Lecture Notes in Computer Science (LNCS)},
volume = {11789},
year = 2019,
eprint = {1902.06103},
archiveprefix = {arXiv},
doi = {10.1007/978-3-030-30786-8_10},
print_isbn = {978-3-030-30785-1},
isbn = {978-3-030-30786-8},
abstract = {Flip graphs are a ubiquitous class of graphs, which encode relations induced on a set of combinatorial objects by elementary, local changes.
A natural computational problem to consider is the flip distance: Given two objects, what is the minimum number of flips needed to transform one into the other?
We consider flip graphs on so-called $\alpha$-orientations of a graph $G$, in which every vertex $v$ has a specified outdegree $\alpha(v)$, and a flip consists of reversing all edges of a directed cycle. We prove that deciding whether the flip distance between two $\alpha$-orientations of a planar graph $G$ is at most 2 is \NP-complete. This also holds in the special case of plane perfect matchings, where flips involve alternating cycles. We also consider the dual question of the flip distance between graph orientations in which every cycle has a specified number of forward edges, and a flip is the reversal of all edges in a minimal directed cut. In general, the problem remains hard, but if we only change sinks into sources, or vice-versa, then the problem can be solved in polynomial time.},
originalfile = {/geometry/cggg.bib}
}
@article{achkmsv-fdbgo-19b,
author = {Oswin Aichholzer and Jean Cardinal and Tony Huynh and Kolja Knauer and Torsten M{\"u}tze and Raphael Steiner and Birgit Vogtenhuber},
title = {{{Flip distances between graph orientations}}},
journal = {Algorithmica},
issn = {0178-4617},
doi = {10.1007/s00453-020-00751-1},
pages = {1--28},
year = 2021,
eprint = {1902.06103},
abstract = {Flip graphs are a ubiquitous class of graphs, which encode relations on a set of combinatorial objects by elementary, local changes.
Skeletons of associahedra, for instance, are the graphs induced by quadrilateral flips in triangulations of a convex polygon.
For some definition of a flip graph, a natural computational problem to consider is the flip distance: Given two objects, what is
the minimum number of flips needed to transform one into the other?
We consider flip graphs on orientations of simple graphs, where flips consist of reversing the direction of some edges.
More precisely, we consider so-called $\alpha$-orientations of a graph $G$, in which every vertex $v$ has a specified outdegree $\alpha(v)$,
and a flip consists of reversing all edges of a directed cycle.
We prove that deciding whether the flip distance between two $\alpha$-orientations of a planar graph $G$ is at most two is \NP-complete.
This also holds in the special case of perfect matchings, where flips involve alternating cycles.
This problem amounts to finding geodesics on the common base polytope of two partition matroids, or, alternatively, on an alcoved polytope.
It therefore provides an interesting example of a flip distance question that is computationally intractable despite having a natural interpretation
as a geodesic on a nicely structured combinatorial polytope.
We also consider the dual question of the flip distance between graph orientations in which every cycle has a specified number of forward edges,
and a flip is the reversal of all edges in a minimal directed cut.
In general, the problem remains hard. However, if we restrict to flips that only change sinks into sources, or vice-versa, then
the problem can be solved in polynomial time. Here we exploit the fact that the flip graph is the cover graph of a distributive lattice.
This generalizes a recent result from Zhang, Qian, and Zhang (Acta. Math. Sin.-English Ser., 2019).},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aacddfkklmms-fphic-19,
author = {Oswin Aichholzer and Hugo A. Akitaya and Kenneth C. Cheung and Erik D. Demaine and Martin L. Demaine and S\'{a}ndor~P.~Fekete and Linda Kleist and Irina Kostitsyna and Maarten L\"offler and Zuzana Mas\'{a}rov\'{a} and Klara Mundilova and Christiane Schmidt},
title = {{{Folding Polyominoes with Holes into a Cube}}},
booktitle = {Proc. $31^{th}$ Annual Canadian Conference on Computational Geometry CCCG 2019},
pages = {164--170},
year = 2019,
address = {Edmonton, Alberta, Canada},
url = {https://sites.ualberta.ca/~cccg2019/cccg2019_proceedings.pdf},
category = {3b},
pdf = {/files/publications/geometry/aacddfkklmms-fphic-19.pdf},
abstract = {When can a polyomino piece of paper be folded into a unit cube?
Prior work studied tree-like polyominoes, but polyominoes with holes remain an
intriguing open problem. We present sufficient conditions for a polyomino with
hole(s) to fold into a cube, and conditions under which cube folding is impossible.
In particular, we show that all but five special simple holes guarantee foldability.},
originalfile = {/geometry/cggg.bib}
}
@article{aacddfkklmms-fphic-21,
title = {{"Folding polyominoes with holes into a cube"}},
journal = {Computational Geometry},
volume = {93},
pages = {101700},
year = {2021},
issn = {0925-7721},
eprint = {1910.09917},
archiveprefix = {arXiv},
pdf = {/files/publications/geometry/aacddfkklmms-fphic-19.pdf},
doi = {https://doi.org/10.1016/j.comgeo.2020.101700},
url = {http://www.sciencedirect.com/science/article/pii/S0925772120300948},
author = {Oswin Aichholzer and Hugo A. Akitaya and Kenneth C. Cheung and Erik D. Demaine and Martin L. Demaine and S{\'a}ndor P. Fekete and Linda Kleist and Irina Kostitsyna and Maarten L{\"o}ffler and Zuzana Mas{\'a}rov{\'a} and Klara Mundilova and Christiane Schmidt},
keywords = {Folding, Origami folding, Cube, Polyomino with holes, Non-simple polyomino},
abstract = {When can a polyomino piece of paper be folded into a unit cube? Prior work studied tree-like polyominoes,
but polyominoes with holes remain an intriguing open problem. We present sufficient conditions for a polyomino
with one or several holes to fold into a cube, and conditions under which cube folding is impossible.
In particular, we show that all but five special `basic'' holes guarantee foldability.},
originalfile = {/geometry/cggg.bib}
}
@article{abhkmppsvvw-mrotg-20,
author = {Oswin Aichholzer and Martin Balko and Michael Hoffmann and Jan Kyn\v{c}l and Wolfgang Mulzer and Irene Parada and Alexander Pilz and Manfred Scheucher and Pavel Valtr and Birgit Vogtenhuber and Emo Welzl},
title = {{{{{Minimal representations of order types by geometric graphs}}}}},
journal = {Journal of Graph Algorithms and Applications},
volume = {24},
number = {4},
pages = {551--572},
doi = {10.7155/jgaa.00545},
note = {special issue of the 27th International Symposium on Graph Drawing and Network Visualization GD$\,$2019},
year = {2020},
abstract = {In order to have a compact visualization of the order type of
a given point set $S$,
we are interested in geometric graphs on $S$ with few edges that unequivocally display
the order type of $S$.
We introduce the concept of \emph{exit edges},
which prevent the order type from changing under continuous motion of vertices.
Exit edges have a natural dual characterization,
which allows us to efficiently compute them and to bound their number.},
originalfile = {/geometry/cggg.bib}
}
@article{adfhg-opirp-19,
author = {Oswin Aichholzer and Frank Duque and Ruy Fabila-Monroy and Carlos Hidalgo-Toscano and Oscar E. Garc\'ia-Quintero},
title = {{{An Ongoing Project to Improve the Rectilinear and the Pseudolinear Crossing Constants}}},
journal = {Journal of Graph Algorithms and Applications},
year = {2020},
volume = {24},
number = {3},
pages = {421--432},
doi = {10.7155/jgaa.00540},
eprint = {1907.07796},
abstract = {A drawing of a graph in the plane is {\it pseudolinear} if the edges of the drawing can be extended to doubly-infinite curves that form an arrangement of pseudolines,
that is, any pair of edges crosses precisely once. A special case are {\it rectilinear} drawings where the edges of the graph are drawn as straight line segments.
The rectilinear (pseudolinear) crossing number of a graph is the minimum number of pairs of edges of the graph that cross in any of its rectilinear (pseudolinear) drawings.
In this paper we describe an ongoing project to continuously obtain better asymptotic upper bounds on the rectilinear and pseudolinear crossing number of the complete graph~$K_n$.},
originalfile = {/geometry/cggg.bib}
}
@article{aabv-tftm-19,
author = {Oswin Aichholzer and Lukas Andritsch and Karin Baur and Birgit Vogtenhuber},
title = {{{Transformed flips in triangulations and matchings}}},
journal = {submitted},
pages = {1--17},
year = 2019,
eprint = {1907.08758},
archiveprefix = {arXiv},
abstract = {Plane perfect matchings of $2n$ points in convex position are in bijection with triangulations of convex polygons of size $n+2$.
Edge flips are a classic operation to perform local changes both structures have in common.
In this work, we use the explicit bijection from Aichholzer et al. (2018) to determine the effect of an edge flip on the one side of the bijection to the other side,
that is, we show how the two different types of edge flips are related. Moreover, we give an algebraic interpretation of the flip graph of triangulations in terms of
elements of the corresponding Temperley-Lieb algebra.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{agpvw-sdcss-20,
author = {Oswin Aichholzer and Alfredo Garc\'ia and Irene Parada and Birgit Vogtenhuber and Alexandra Weinberger},
title = {{{Simple Drawings of {$K_{m,n}$} Contain Shooting Stars}}},
booktitle = {Proceedings of the {36th} European Workshop on Computational Geometry (EuroCG$\,$2020)},
pages = {36:1--36:7},
year = 2020,
address = {W\"urzburg, Germany},
pages = {},
url = {http://www1.pub.informatik.uni-wuerzburg.de/eurocg2020/data/uploads/papers/eurocg20_paper_36.pdf},
abstract = {Simple drawings are drawings of graphs in which all edges have at most one common point
(either a common endpoint, or a proper crossing).
It has been an open question whether every simple drawing of a complete bipartite graph
$K_{m,n}$ contains a plane spanning tree as a subdrawing.
We answer this question to the positive by showing that for every simple drawing of $K_{m,n}$
and for every vertex $v$ in that drawing, the drawing contains a \emph{shooting star}
rooted at $v$, that is, a plane spanning tree with all incident edges of $v$.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{acdfpvv-sdcoe-20,
author = {O. Aichholzer and L. E. Caraballo and J.M. D\'iaz-B\'a\~nez and R. Fabila-Monroy and I. Parada and I. Ventura and B. Vogtenhuber},
title = {{{Scheduling drones to cover outdoor events}}},
booktitle = {Proceedings of the {36th} European Workshop on Computational Geometry (EuroCG$\,$2020)},
pages = {24:1--24:7},
year = 2020,
address = {W\"urzburg, Germany},
pages = {},
url = {http://www1.pub.informatik.uni-wuerzburg.de/eurocg2020/data/uploads/papers/eurocg20_paper_24.pdf},
abstract = {Task allocation is an important aspect of many multi-robot systems.
In this paper, we consider a new task allocation problem that appears
in multi-robot aerial cinematography. The main goal is to distribute a
set of tasks (shooting actions) among the team members optimizing a
certain objective function.
The tasks are given as sequences of waypoints with associated time
intervals (scenes). We prove that the task allocation problem maximizing
the total filmed time by $k$ aerial robots (drones) can be solved in
polynomial time when the drones do not require battery recharge. We also
consider the problem in which the drones have a limited battery endurance
and must periodically go to a static base station. For this version, we
show how to solve the problem in polynomial time when only one drone is
available.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{abbcfmv-dgs-20,
author = {Oswin Aichholzer and Manuel Borrazzo and Prosenjit Bose and Jean Cardinal and Fabrizio Frati and Pat Morin and Birgit Vogtenhuber},
title = {{{Drawing Graphs as Spanners}}},
booktitle = {45th International Workshop on Graph-Theoretic Concepts in Computer Science. WG$\,$2020.},
__booktitle = {Graph-Theoretic Concepts in Computer Science. WG$\,$2020.},
editor = {Adler, Isolde and M{\"u}ller, Haiko},
address = {Leeds, United Kingdom},
publisher = {Springer International Publishing},
series = {Lecture Notes in Computer Science (LNCS)},
volume = {12301},
pages = {310--324},
year = 2020,
isbn = {978-3-030-60440-0},
doi = {10.1007/978-3-030-60440-0_25},
eprint = {2002.05580},
archiveprefix = {arXiv},
pdf = {/files/publications/geometry/abbcfmv-dgs-20.pdf},
abstract = {We study the problem of embedding graphs in the plane as good geometric spanners. That is, for a graph $G$, the goal is to construct a straight-line drawing $\Gamma$ of $G$ in the plane such that, for any two vertices $u$ and $v$ of $G$, the ratio between the minimum length of any path from $u$ to $v$ and the Euclidean distance between $u$ and $v$ is small. The maximum such ratio, over all pairs of vertices of~$G$, is the \emph{spanning ratio} of $\Gamma$.
First, we show that deciding whether a graph admits a straight-line drawing with spanning ratio~$1$, a proper straight-line drawing with spanning ratio~$1$, and a planar straight-line drawing with spanning ratio~$1$ are NP-complete, $\exists \mathbb R$-complete, and linear-time solvable problems, respectively.
Second, we prove that, for every $\epsilon>0$, every (planar) graph admits a proper (resp.\ planar) straight-line drawing with spanning ratio smaller than~$1+\epsilon$.
Third, we note that our drawings with spanning ratio smaller than~$1+\epsilon$ have large edge-length ratio, that is, the ratio between the lengths of the longest and of the shortest edge is exponential. We show that this is sometimes unavoidable. More generally, we identify having bounded toughness as the criterion that distinguishes graphs that admit straight-line drawings with constant spanning ratio and polynomial edge-length ratio from graphs that do not.},
originalfile = {/geometry/cggg.bib}
}
@article{abbcfmv-dgs-22,
author = {Aichholzer, Oswin and Borrazzo, Manuel and Bose, Prosenjit and Cardinal, Jean and Frati, Fabrizio and Morin, Pat and Vogtenhuber, Birgit},
year = {2022},
month = {06},
pages = {774--795},
volume = {68},
title = {{{Drawing Graphs as Spanners}}},
journal = {Discrete & Computational Geometry},
eprint = {2002.05580},
archiveprefix = {arXiv},
doi = {10.1007/s00454-022-00398-5},
pdf = {/files/publications/geometry/abbcfmv-dgs-20.pdf},
abstract = {We study the problem of embedding graphs in the plane as good geometric spanners. That is, for a graph $G$, the goal is to construct a straight-line drawing $\Gamma$ of $G$ in the plane such that, for any two vertices $u$ and $v$ of $G$, the ratio between the minimum length of any path from $u$ to $v$ and the Euclidean distance between $u$ and $v$ is small. The maximum such ratio, over all pairs of vertices of~$G$, is the \emph{spanning ratio} of $\Gamma$.
First, we show that deciding whether a graph admits a straight-line drawing with spanning ratio~$1$, a proper straight-line drawing with spanning ratio~$1$, and a planar straight-line drawing with spanning ratio~$1$ are NP-complete, $\exists \mathbb R$-complete, and linear-time solvable problems, respectively.
Second, we prove that, for every $\epsilon>0$, every (planar) graph admits a proper (resp.\ planar) straight-line drawing with spanning ratio smaller than~$1+\epsilon$.
Third, we note that our drawings with spanning ratio smaller than~$1+\epsilon$ have large edge-length ratio, that is, the ratio between the lengths of the longest and of the shortest edge is exponential. We show that this is sometimes unavoidable. More generally, we identify having bounded toughness as the criterion that distinguishes graphs that admit straight-line drawings with constant spanning ratio and polynomial edge-length ratio from graphs that do not.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aoppt-dtcppm-20,
author = {O. Aichholzer and J. Obmann and P. Pat\'ak and D. Perz and J. Tkadlec},
title = {{{Disjoint tree-compatible plane perfect matchings}}},
booktitle = {Proceedings of the {36th} European Workshop on Computational Geometry (EuroCG$\,$2020)},
pages = {56:1--56:7},
year = 2020,
address = {W\"urzburg, Germany},
pages = {},
url = {http://www1.pub.informatik.uni-wuerzburg.de/eurocg2020/data/uploads/papers/eurocg20_paper_56.pdf},
abstract = {Two plane drawings of geometric graphs on the same set of points are called disjoint compatible if their union is plane and they do not have an edge in common.
For a given set $S$ of $2n$ points two plane drawings of perfect matchings $M_1$ and $M_2$ (which do not need to be disjoint nor compatible) are \emph{disjoint tree-compatible} if there exists a plane drawing of a spanning tree $T$ on~$S$ which is disjoint compatible to both $M_1$ and $M_2$.
We show that the graph of all disjoint tree-compatible perfect geometric matchings on $2n$ points in convex position is connected if and only if $2n \geq 10$. Moreover, in that case the diameter of this graph is either 4 or 5, independent of $n$.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aamppptv-cm2021,
author = {Oswin Aichholzer and Alan Arroyo and Zuzana Mas\'{a}rov\'{a} and Irene Parada and Daniel Perz and Alexander Pilz and Josef Tkadlec and Birgit Vogtenhuber},
title = {{{On Compatible Matchings}}},
booktitle = {WALCOM: Algorithms and Computation},
year = {2021},
publisher = {Springer International Publishing},
adress = {Cham},
pages = {221--233},
editor = {Ryuhei Uehara and Seok-Hee Hong and Subhas C. Nandy},
eprint = {2101.03928},
doi = {10.1007/978-3-030-68211-8_18},
note = {Best Paper Award},
isbn = {978-3-030-68211-8},
abstract = {A matching is compatible to two or more labeled point sets of size $n$ with labels $\{1,\dots,n\}$ if its straight-line drawing on each of these point sets is crossing-free. We study the maximum number of edges in a matching compatible to two or more labeled point sets in general position in the plane. We show that for any two labeled convex sets of $n$ points there exists a compatible matching with $\lfloor \sqrt {2n}\rfloor$ edges. More generally, for any $\ell$ labeled point sets we construct compatible matchings of size $\Omega(n^{1/\ell})$. As a corresponding upper bound, we use probabilistic arguments to show that for any $\ell$ given sets of $n$ points there exists a labeling of each set such that the largest compatible matching has $\mathcal{O}(n^{2/(\ell+1)})$ edges. Finally, we show that $\Theta(\log n)$ copies of any set of $n$ points are necessary and sufficient for the existence of a labeling such that any compatible matching consists only of a single edge.},
originalfile = {/geometry/cggg.bib}
}
@article{aamppptv-cm2022,
author = {{Oswin} {Aichholzer} and {Alan} {Arroyo} and {Zuzana} {Mas\'{a}rov\'{a}} and {Irene} {Parada} and {Daniel} {Perz} and {Alexander} {Pilz} and {Josef} {Tkadlec} and {Birgit} {Vogtenhuber}},
title = {{{On Compatible Matchings}}},
journal = {Journal of Graph Algorithms and Applications},
year = {2022},
volume = {26},
number = {2},
pages = {225--240},
doi = {10.7155/jgaa.00591},
abstract = {A matching is compatible to two or more labeled point sets of size $n$ with labels $\{1,\dots,n\}$ if its straight-line drawing on each of these point sets is crossing-free. We study the maximum number of edges in a matching compatible to two or more labeled point sets in general position in the plane. We show that for any two labeled convex sets of $n$ points there exists a compatible matching with $\lfloor \sqrt {2n}\rfloor$ edges. More generally, for any $\ell$ labeled point sets we construct compatible matchings of size $\Omega(n^{1/\ell})$. As a corresponding upper bound, we use probabilistic arguments to show that for any $\ell$ given sets of $n$ points there exists a labeling of each set such that the largest compatible matching has $\mathcal{O}(n^{2/(\ell+1)})$ edges. Finally, we show that $\Theta(\log n)$ copies of any set of $n$ points are necessary and sufficient for the existence of a labeling such that any compatible matching consists only of a single edge.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ahoppsvw-pstec20,
author = {Oswin Aichholzer and Michael Hoffmann and Johannes Obenaus and Rosna Paul and Daniel Perz and Nadja Seiferth and Birgit Vogtenhuber and Alexandra Weinberger},
title = {{{Plane Spanning Trees in Edge-Colored Simple Drawings of $K_n$}}},
booktitle = {Graph Drawing and Network Visualization (GD 2020)},
series = {Lecture Notes in Computer Science (LNCS)},
nonote = {28th International Symposium on Graph Drawing and Network Visualization},
doi = {10.1007/978-3-030-68766-3_37},
url = {https://doi.org/10.1007%2F978-3-030-68766-3_37},
publisher = {Springer International Publishing},
pages = {482--489},
eprint = {2008.08827},
archiveprefix = {arXiv},
abstract = {K{\'{a}}rolyi, Pach, and T{\'{o}}th proved that every 2-edge-colored
straight-line drawing of the complete graph contains a monochromatic plane spanning tree.
It is open if this statement generalizes to other classes of drawings, specifically, to \emph{simple drawings} of the complete graph.
These are drawings where edges are represented by Jordan arcs, any two of which intersect at most once.
We present two partial results towards such a generalization. First, we show that the statement holds for cylindrical simple drawings.
(In a \emph{cylindrical} drawing, all vertices are placed on two concentric circles and no edge crosses either circle.)
Second, we introduce a relaxation of the problem in which the graph is $k$-edge-colored,
and the target structure must be \emph{hypochromatic}, that is, avoid (at least) one color class.
In this setting, we show that every $\lceil (n+5)/6\rceil$-edge-colored
monotone simple drawing of $K_n$ contains a hypochromatic plane spanning tree.
(In a \emph{monotone} drawing, every edge is represented as an $x$-monotone curve.)},
year = {2021},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aksv-o4cfpsaub-21,
author = {Oswin Aichholzer and Jan Kyn\v{c}l and Manfred Scheucher and Birgit Vogtenhuber},
title = {{{On 4-Crossing-Families in Point Sets and an Asymptotic Upper Bound}}},
booktitle = {Proceedings of the {37th} European Workshop on Computational Geometry (EuroCG$\,$2021)},
year = 2021,
address = {St. Petersburg, Germany},
pages = {38:1--38:8},
eprint = {2109.10705},
archiveprefix = {arXiv},
archiveprefix = {arXiv},
url = {http://eurocg21.spbu.ru/wp-content/uploads/2021/04/EuroCG_2021_paper_38.pdf},
originalfile = {/geometry/cggg.bib}
}
@article{aksvv-cfps-22,
title = {{{On crossing-families in planar point sets}}},
journal = {Computational Geometry},
volume = {107},
pages = {101-899},
year = {2022},
issn = {0925-7721},
eprint = {2109.10705},
archiveprefix = {arXiv},
doi = {https://doi.org/10.1016/j.comgeo.2022.101899},
url = {https://www.sciencedirect.com/science/article/pii/S0925772122000426},
author = {Oswin Aichholzer and Jan Kyn\v{c}l and Manfred Scheucher and Birgit Vogtenhuber and Pavel Valtr},
keywords = {Crossing family, Point set, Order type, Geometric thrackle},
abstract = {A k-crossing family in a point set S in general position is a set of k segments spanned by points of S such that all k segments mutually cross. In this short note we present two statements on crossing families which are based on sets of small cardinality: (1) Any set of at least 15 points contains a crossing family of size 4. (2) There are sets of n points which do not contain a crossing family of size larger than 8⌈n41⌉. Both results improve the previously best known bounds.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aeh-gdsmvntilp-21,
author = {Oswin Aichholzer and David Eppstein and Eva-Maria Hainzl},
title = {{{Geometric Dominating Sets -- A Minimum Version of the No-Three-In-Line Problem}}},
booktitle = {Proceedings of the {37th} European Workshop on Computational Geometry (EuroCG$\,$2021)},
year = 2021,
address = {St. Petersburg, Germany},
pages = {17:1--17:7},
eprint = {2203.13170},
archiveprefix = {arXiv},
url = {http://eurocg21.spbu.ru/wp-content/uploads/2021/04/EuroCG_2021_paper_17.pdf},
abstract = {We consider a minimizing variant of the well-known \emph{No-Three-In-Line Problem}, the \emph{Geometric Dominating Set Problem}: What is the smallest number of points in an $n\times n$~grid such that every grid point lies on a common line with two of the points in the set? We show a lower bound of $\Omega(n^{2/3})$ points and provide a constructive upper bound of size $2 \lceil n/2 \rceil$. If the points of the dominating sets are required to be in general position we provide optimal solutions for grids of size up to $12 \times 12$. For arbitrary $n$ the currently best upper bound remains the obvious $2n$. Finally, we discuss some further variations of the problem.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{agtvw-pmsdcg-21,
author = {Oswin Aichholzer and Alfredo Garc\'{\i}a and Javier Tejel and Birgit Vogtenhuber and Alexandra Weinberger},
title = {{{Plane Matchings in Simple Drawings of Complete Graphs}}},
booktitle = {Proceedings of the Computational Geometry: Young Researchers Forum},
year = {2021},
pages = {6-10},
url = {https://cse.buffalo.edu/socg21/files/YRF-Booklet.pdf#page=6},
abstract = {Simple drawings are drawings of graphs in which the edges are Jordan arcs and each pair of edges shares at most one point (a proper crossing or a common endpoint). We show that every simple drawing of the complete graph with~$n$ vertices contains~$\Omega(n^{\frac{1}{2}})$ pairwise disjoint edges. This improves the currently known best lower bound $\Omega(n^{\frac{1}{2}-\varepsilon})$ for any $\varepsilon>0$ by Ruiz-Vargas.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{afkpppsv-pmwc-21,
author = {Oswin Aichholzer and Ruy Fabila-Monroy and Philipp Kindermann and Irene Parada and Rosna Paul and Daniel Perz and Patrick Schnider and Birgit Vogtenhuber},
title = {{{Perfect Matchings with Crossings}}},
booktitle = {Proceedings of the Computational Geometry: Young Researchers Forum},
year = {2021},
pages = {24-27},
url = {https://cse.buffalo.edu/socg21/files/YRF-Booklet.pdf#page=24},
abstract = {In this paper, we analyze the number of straight-line perfect matchings with $k$ crossings on point sets of size $n$ = $2m$ in general position. We show that for every $k\leq 5n/8-\Theta(1)$, every $n$-point set admits a perfect matching with exactly $k$ crossings and that there exist $n$-point sets where every perfect matching has fewer than $5n^2/72$ crossings. We also study the number of perfect matchings with at most $k$ crossings. Finally we show that convex point sets %in convex position maximize the number of perfect matchings with $n/2 \choose 2$ crossings and ${n/2 \choose 2}\!-\!1$ crossings.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{agtvw-pmsdcg-21b,
author = {Oswin Aichholzer and Alfredo Garc\'{\i}a and Javier Tejel and Birgit Vogtenhuber and Alexandra Weinberger},
title = {{{Plane paths in simple drawings of complete graphs}}},
booktitle = {Proc. XIX Encuentros de Geometr\'{\i}a Computacional},
year = {2021},
pages = {4},
abstract = {Simple drawings are drawings of graphs in the plane such that vertices are distinct points in the plane, edges are Jordan arcs connecting their endpoints, and edges intersect at most once either in a proper crossing or in a shared endpoint. It is conjectured that every simple drawing of the complete graph with $n$ vertices, $K_n$, contains a plane Hamiltonian cycle, and consequently a plane Hamiltonian path.
However, to the best of our knowledge, $\Omega((\log n)^{1/6})$ is currently the best known lower bound for the length of a plane path contained in any simple drawing of $K_n$. We improve this bound to $\Omega(\log n / (\log \log n) )$.},
url = {https://quantum-explore.com/wp-content/uploads/2021/06/Actas_egc21.pdf#page=11},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{allmopppvw-d-21,
author = {Aichholzer, Oswin and L{\"o}ffler, Maarten and Lynch, Jayson and Mas{\'a}rov{\'a}, Zuzana and Orthaber, Joachim and Parada, Irene and Paul, Rosna and Perz, Daniel and Vogtenhuber, Birgit and Weinberger, Alexandra},
title = {{{Dominect: A Simple yet Deep 2-Player Board Game}}},
year = {2021},
pages = {112--113},
booktitle = {23rd Thailand-Japan Conference on Discrete and Computational Geometry, Graphs, and Games (TJCDCGGG 2020+1)},
url = {http://www.math.science.cmu.ac.th/tjcdcggg/},
abstract = {In this work we introduce the perfect information 2-player game \emph{Dominect}, which has recently been invented by two of the authors. Despite being a game with quite simple rules, Dominect reveals a high depth of complexity. We report on first results concerning the development of winning strategies, as well as a PSPACE-hardness result for deciding whether a given game position is a winning position.},
originalfile = {/geometry/cggg.bib}
}
@article{aichholzer2021edge,
title = {{{Edge Partitions of Complete Geometric Graphs (Part 2)}}},
author = {Oswin Aichholzer and Johannes Obenaus and Joachim Orthaber and Rosna Paul and Patrick Schnider and Raphael Steiner and Tim Taubner and Birgit Vogtenhuber},
year = {2021},
eprint = {2112.08456},
archiveprefix = {arXiv},
abstract = {Recently, the second and third author showed that complete geometric graphs on $2n$ vertices in general cannot be partitioned into $n$ plane spanning trees. Building up on this work, in this paper, we initiate the study of partitioning into beyond planar subgraphs, namely into $k$-planar and $k$-quasi-planar subgraphs and obtain first bounds on the number of subgraphs required in this setting.},
primaryclass = {math.CO},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{adkllmrvw-htst-22,
author = {Aichholzer, Oswin and Demaine, Erik D. and Korman, Matias and Lubiw, Anna and Lynch, Jayson and Mas\'{a}rov\'{a}, Zuzana and Rudoy, Mikhail and Vassilevska Williams, Virginia and Wein, Nicole},
title = {{{{Hardness of Token Swapping on Trees}}}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {3:1--3:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
isbn = {978-3-95977-247-1},
issn = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
url = {https://drops.dagstuhl.de/opus/volltexte/2022/16941},
urn = {urn:nbn:de:0030-drops-169413},
doi = {10.4230/LIPIcs.ESA.2022.3},
eprint = {2103.06707},
archiveprefix = {arXiv},
abstract = {Given a graph where every vertex has exactly one labeled token, how can we most quickly execute a given permutation on the tokens? In (sequential) token swapping, the goal is to use the shortest possible sequence of swaps, each of which exchanges the tokens at the two endpoints of an edge of the graph. In parallel token swapping, the goal is to use the fewest rounds, each of which consists of one or more swaps on the edges of a matching. We prove that both of these problems remain NP-hard when the graph is restricted to be a tree. These token swapping problems have been studied by disparate groups of researchers in discrete mathematics, theoretical computer science, robot motion planning, game theory, and engineering. Previous work establishes NP-completeness on general graphs (for both problems); polynomial-time algorithms for simple graph classes such as cliques, stars, paths, and cycles; and constant-factor approximation algorithms in some cases. The two natural cases of sequential and parallel token swapping in trees were first studied over thirty years ago (as "sorting with a transposition tree") and over twenty-five years ago (as "routing permutations via matchings"), yet their complexities were previously unknown. We also show limitations on approximation of sequential token swapping on trees: we identify a broad class of algorithms that encompass all three known polynomial-time algorithms that achieve the best known approximation factor (which is $2$) and show that no such algorithm can achieve an approximation factor less than $2$.},
annote = {Keywords: Sorting, Token swapping, Trees, NP-hard, Approximation},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{achhmv-gtcbg-22,
author = {Oswin Aichholzer and Man-Kwun Chiu and Hung P. Hoang and Michael Hoffmann and
Yannic Maus and Birgit Vogtenhuber and Alexandra Weinberger},
title = {{{Gioan{'}s Theorem for complete bipartite graphs}}},
booktitle = {Proc. $38^{th}$ European Workshop on Computational Geometry (EuroCG 2022)},
pages = {31:1--31:6},
year = 2022,
address = {Perugia, Italy},
pdf = {https://eurocg2022.unipg.it/booklet/EuroCG2022-Booklet.pdf#page=226},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{ahlppsv-bdte-22,
author = {Oswin Aichholzer and Thomas Hackl and Maarten L{\"o}ffler and Alexander
Pilz and Irene Parada and Manfred Scheucher and Birgit Vogtenhuber},
title = {{{Blocking Delaunay Triangulations from Exterior}}},
booktitle = {Proc. $38^{th}$ European Workshop on Computational Geometry (EuroCG 2022)},
pages = {9:1--9:7},
year = 2022,
address = {Perugia, Italy},
eprint = {2210.12015},
archiveprefix = {arXiv},
pdf = {https://eurocg2022.unipg.it/booklet/EuroCG2022-Booklet.pdf#page=65},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aklmmopv-fpsp-22,
author = {Oswin Aichholzer and Kristin Knorr and Maarten L{\"o}ffler and Zuzana Mas{\'{a}}rov{\'{a}} and Wolfgang Mulzer and Johannes Obenaus and Rosna Paul and Birgit Vogtenhuber},
title = {{{Flipping Plane Spanning Paths}}},
booktitle = {Proc. $38^{th}$ European Workshop on Computational Geometry (EuroCG 2022)},
pages = {66:1--66:7},
year = 2022,
address = {Perugia, Italy},
eprint = {2202.10831},
archiveprefix = {arXiv},
pdf = {https://eurocg2022.unipg.it/booklet/EuroCG2022-Booklet.pdf#page=478},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{agtvw-twfpssdcg-22,
author = {Oswin Aichholzer and Alfredo Garc\'{\i}a and Javier Tejel and Birgit Vogtenhuber and Alexandra Weinberger},
title = {{{{Twisted Ways to Find Plane Structures in Simple Drawings of Complete Graphs}}}},
booktitle = {38th International Symposium on Computational Geometry
(SoCG 2022)},
pages = {5:1--5:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
isbn = {978-3-95977-227-3},
issn = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
eprint = {2203.06143},
archiveprefix = {arXiv},
url = {https://drops.dagstuhl.de/opus/volltexte/2022/16013},
doi = {10.4230/LIPIcs.SoCG.2022.5},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aoopsstv-epcgg-22,
author = {Aichholzer, Oswin and Obenaus, Johannes and Orthaber, Joachim
and Paul, Rosna and Schnider, Patrick and Steiner, Raphael and Taubner,
Tim and Vogtenhuber, Birgit},
title = {{{{Edge Partitions of Complete Geometric Graphs}}}},
booktitle = {38th International Symposium on Computational Geometry
(SoCG 2022)},
pages = {6:1--6:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
isbn = {978-3-95977-227-3},
issn = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
url = {https://drops.dagstuhl.de/opus/volltexte/2022/16014},
urn = {urn:nbn:de:0030-drops-160141},
doi = {10.4230/LIPIcs.SoCG.2022.6},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{afkpppsv-pmc-22,
author = {Aichholzer, Oswin and Fabila-Monroy, Ruy and Kindermann, Philipp and Parada, Irene and Paul, Rosna and Perz, Daniel and Schnider, Patrick and Vogtenhuber, Birgit},
title = {{{Perfect Matchings with Crossings}}},
year = {2022},
isbn = {978-3-031-06677-1},
publisher = {Springer-Verlag},
address = {Berlin, Heidelberg},
doi = {10.1007/978-3-031-06678-8_4},
booktitle = {Combinatorial Algorithms: 33rd International Workshop, IWOCA 2022, Trier, Germany, June 7--9, 2022, Proceedings},
pages = {46--59},
numpages = {14},
keywords = {Combinatorial geometry, Order types, Perfect matchings, Crossings},
location = {Trier, Germany},
originalfile = {/geometry/cggg.bib}
}
@article{aeh-gdsmv-22,
title = {{{Geometric dominating sets - a minimum version of the No-Three-In-Line Problem}}},
journal = {Computational Geometry},
volume = {108},
pages = {101--913},
year = {2023},
issn = {0925-7721},
eprint = {2203.13170},
archiveprefix = {arXiv},
doi = {https://doi.org/10.1016/j.comgeo.2022.101913},
url = {https://www.sciencedirect.com/science/article/pii/S0925772122000566},
author = {Oswin Aichholzer and David Eppstein and Eva-Maria Hainzl},
keywords = {No-Three-In-Line Problem, Point sets in general position, Dominating sets, Domination number, Geometric domination},
abstract = {We consider a minimizing variant of the well-known No-Three-In-Line Problem, the Geometric Dominating Set Problem: What is the smallest number of points in an n×n grid such that every grid point lies on a common line with two of the points in the set? We show a lower bound of Ω(n2/3) points and provide a constructive upper bound of size 2⌈n/2⌉. If the points of the dominating sets are required to be in general position we provide optimal solutions for grids of size up to 12×12. For arbitrary n the currently best upper bound for points in general position remains the obvious 2n. Finally, we discuss the problem on the discrete torus where we prove an upper bound of O((nlogn)1/2). For n even or a multiple of 3, we can even show a constant upper bound of 4. We also mention a number of open questions and some further variations of the problem.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{a-aslscn-21,
title = {{{Another Small but Long Step for Crossing Numbers: {$cr(13)}} = 225$ and $cr(14) = 315$}},
author = {Aichholzer, Oswin},
booktitle = {Proceedings of the 33rd Canadian Conference on Computational Geometry (CCCG 2021)},
pages = {72--77},
year = {2021},
url = {https://projects.cs.dal.ca/cccg2021/proceedings/},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{akmeoprvw-cstsd-23,
author = {Aichholzer, Oswin
and Knorr, Kristin
and Mulzer, Wolfgang
and El Maalouly, Nicolas
and Obenaus, Johannes
and Paul, Rosna
and M. Reddy, Meghana
and Vogtenhuber, Birgit
and Weinberger, Alexandra},
editor = {Angelini, Patrizio and von Hanxleden, Reinhard},
title = {{{Compatible Spanning Trees in Simple Drawings of {$K_{n}$}}}},
booktitle = {Graph Drawing and Network Visualization},
year = {2023},
publisher = {Springer International Publishing},
address = {Cham},
pages = {16--24},
isbn = {978-3-031-22203-0},
doi = {10.1007/978-3-031-22203-0_2},
eprint = {2208.11875},
archiveprefix = {arXiv},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{agpvw-sssd-23,
author = {Aichholzer, Oswin
and Garc{\'i}a, Alfredo
and Parada, Irene
and Vogtenhuber, Birgit
and Weinberger, Alexandra},
editor = {Angelini, Patrizio and von Hanxleden, Reinhard},
title = {{{Shooting Stars in Simple Drawings of {$K_{m,n}$}}}},
booktitle = {Graph Drawing and Network Visualization},
year = {2023},
publisher = {Springer International Publishing},
address = {Cham},
pages = {49--57},
isbn = {978-3-031-22203-0},
doi = {10.1007/978-3-031-22203-0_5},
eprint = {2209.01190},
archiveprefix = {arXiv},
originalfile = {/geometry/cggg.bib}
}
@article{akoppvv-gwlta-23,
title = {{{Graphs with large total angular resolution}}},
journal = {Theoretical Computer Science},
volume = {943},
pages = {73-88},
year = {2023},
issn = {0304-3975},
eprint = {1908.06504},
archiveprefix = {arXiv},
doi = {https://doi.org/10.1016/j.tcs.2022.12.010},
url = {https://www.sciencedirect.com/science/article/pii/S0304397522007320},
author = {Oswin Aichholzer and Matias Korman and Yoshio Okamoto and Irene Parada and Daniel Perz and Andr\'e {van Renssen} and Birgit Vogtenhuber},
keywords = {Graph drawing, Total angular resolution, Angular resolution, Crossing resolution, NP-hardness},
abstract = {The total angular resolution of a straight-line drawing is the minimum angle between two edges of the drawing. It combines two properties contributing to the readability of a drawing: the angular resolution, which is the minimum angle between incident edges, and the crossing resolution, which is the minimum angle between crossing edges. We consider the total angular resolution of a graph, which is the maximum total angular resolution of a straight-line drawing of this graph. We prove tight bounds for the number of edges for graphs for some values of the total angular resolution up to a finite number of well specified exceptions of constant size. In addition, we show that deciding whether a graph has total angular resolution at least 60∘ is NP-hard. Further we present some special graphs and their total angular resolution.},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aopptv-dcvgc-22,
author = {Aichholzer, Oswin and Obmann, Julia and Pat{\'a}k, Pavel and Perz, Daniel and Tkadlec, Josef and Vogtenhuber, Birgit},
editor = {Bekos, Michael A. and Kaufmann, Michael},
title = {{{Disjoint Compatibility via Graph Classes}}},
booktitle = {Graph-Theoretic Concepts in Computer Science. WG 2022.},
year = {2022},
publisher = {Springer International Publishing},
address = {Cham},
pages = {16--28},
doi = {https://doi.org/10.1007/978-3-031-15914-5_2},
series = {Lecture Notes in Computer Science (LNCS)},
volume = {13453},
abstract = {Two plane drawings of graphs on the same set of points are called disjoint compatible if their union is plane and they do not have an edge in common.
Let $S$ be a convex point set of $2n \geq 10$ points and let $\mathcal{H}$ be a family of plane drawings on $S$.
Two plane perfect matchings $M_1$ and $M_2$ on $S$ (which do not need to be disjoint nor compatible) are \emph{disjoint $\mathcal{H}$-compatible} if there exists a drawing in $\mathcal{H}$ which is disjoint compatible to both $M_1$ and $M_2$.
In this work, we consider the graph which has all plane perfect matchings as vertices and where two vertices are connected by an edge if the matchings are disjoint $\mathcal{H}$-compatible.
We study the diameter of this graph when $\mathcal{H}$ is the family of all plane spanning trees, caterpillars or paths.
We show that in the first two cases the graph is connected with constant and linear diameter, respectively, while in the third case it is disconnected.},
isbn = {978-3-031-15914-5},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aov-2023-tcfhcsdcg,
title = {{{Towards Crossing-Free Hamiltonian Cycles in Simple Drawings of Complete Graphs}}},
author = {Aichholzer, Oswin and Orthaber, Joachim and Vogtenhuber, Birgit},
booktitle = {Proceedings of the 39th European Workshop on Computational Geometry (EuroCG 2023)},
pages = {33:1--33:7},
year = {2023},
url = {https://dccg.upc.edu/eurocg23/wp-content/uploads/2023/03/Session-5B-Talk-2.pdf},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{terbo-ab-23,
title = {{{Two Equivalent Representations of Bicolored Order Types}}},
author = {Oswin Aichholzer and Anna Br\"otzner},
booktitle = {Proceedings of the 39th European Workshop on Computational Geometry (EuroCG 2023)},
pages = {1:1--1:6},
year = {2023},
url = {https://dccg.upc.edu/eurocg23/wp-content/uploads/2023/05/Booklet_EuroCG2023.pdf},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{bpmc-afpsv-23,
title = {{{Bichromatic Perfect Matchings with Crossings}}},
author = {Oswin Aichholzer and Stefan Felsner and Rosna Paul and Manfred Scheucher and Birgit Vogtenhuber},
booktitle = {Proceedings of the 39th European Workshop on Computational Geometry (EuroCG 2023)},
pages = {28:1--28:7},
year = {2023},
url = {https://dccg.upc.edu/eurocg23/wp-content/uploads/2023/05/Booklet_EuroCG2023.pdf},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{akmopv-fpsp-23,
author = {Aichholzer, Oswin and Knorr, Kristin and Mulzer, Wolfgang and Obenaus, Johannes and Paul, Rosna and Vogtenhuber, Birgit},
year = {2023},
month = {03},
pages = {49-60},
title = {{{Flipping Plane Spanning Paths}}},
isbn = {978-3-031-27050-5},
doi = {10.1007/978-3-031-27051-2_5},
editor = {Lin, Chun-Cheng and Lin, Bertrand M. T. and Liotta, Giuseppe},
booktitle = {WALCOM: Algorithms and Computation},
publisher = {Springer Nature Switzerland},
address = {Cham},
abstract = {Let S be a planar point set in general position, and let {\$}{\$}{\backslash}mathcal {\{}P{\}}(S){\$}{\$}be the set of all plane straight-line paths with vertex set S. A flip on a path {\$}{\$}P {\backslash}in {\backslash}mathcal {\{}P{\}}(S){\$}{\$}is the operation of replacing an edge e of P with another edge f on S to obtain a new valid path from {\$}{\$}{\backslash}mathcal {\{}P{\}}(S){\$}{\$}. It is a long-standing open question whether for every given point set S, every path from {\$}{\$}{\backslash}mathcal {\{}P{\}}(S){\$}{\$}can be transformed into any other path from {\$}{\$}{\backslash}mathcal {\{}P{\}}(S){\$}{\$}by a sequence of flips. To achieve a better understanding of this question, we show that it is sufficient to prove the statement for plane spanning paths whose first edge is fixed. Furthermore, we provide positive answers for special classes of point sets, namely, for wheel sets and generalized double circles (which include, e.g., double chains and double circles).},
eprint = {2202.10831},
archiveprefix = {arXiv},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{aichholzer_et_al:LIPIcs.SoCG.2023.6,
author = {Aichholzer, Oswin and Chiu, Man-Kwun and Hoang, Hung P. and Hoffmann, Michael and Kyn\v{c}l, Jan and Maus, Yannic and Vogtenhuber, Birgit and Weinberger, Alexandra},
title = {{{{Drawings of Complete Multipartite Graphs up to Triangle Flips}}}},
booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)},
pages = {6:1--6:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
isbn = {978-3-95977-273-0},
issn = {1868-8969},
year = {2023},
volume = {258},
editor = {Chambers, Erin W. and Gudmundsson, Joachim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
url = {https://drops.dagstuhl.de/opus/volltexte/2023/17856},
urn = {urn:nbn:de:0030-drops-178563},
doi = {10.4230/LIPIcs.SoCG.2023.6},
annote = {Keywords: Simple drawings, simple topological graphs, complete graphs, multipartite graphs, k-partite graphs, bipartite graphs, Gioan’s Theorem, triangle flips, Reidemeister moves},
originalfile = {/geometry/cggg.bib}
}
@article{twisted,
author = {Aichholzer, Oswin and Garc{\'i}a, Alfredo and Tejel, Javier and Vogtenhuber, Birgit and Weinberger, Alexandra},
year = {2024},
title = {{{Twisted Ways to Find Plane Structures in Simple Drawings of Complete Graphs}}},
journal = {Discrete \& Computational Geometry},
pages = {40-66},
volume = {71},
abstract = {Simple drawings are drawings of graphs in which the edges are Jordan arcs and each pair of edges share at most one point (a proper crossing or a common endpoint). A simple drawing is c-monotone if there is a point O such that each ray emanating from O crosses each edge of the drawing at most once. We introduce a special kind of c-monotone drawings that we call generalized twisted drawings. A c-monotone drawing is generalized twisted if there is a ray emanating from O that crosses all the edges of the drawing. Via this class of drawings, we show that every simple drawing of the complete graph with n vertices contains $$\Omega (n^{\frac{1}{2}})$$pairwise disjoint edges and a plane cycle (and hence path) of length $$\Omega (\frac{\log n }{\log \log n})$$. Both results improve over best previously published lower bounds. On the way we show several structural results and properties of generalized twisted and c-monotone drawings, some of which we believe to be of independent interest. For example, we show that a drawing D is c-monotone if there exists a point O such that no edge of D is crossed more than once by any ray that emanates from O and passes through a vertex of D.},
doi = {10.1007/s00454-023-00610-0},
originalfile = {/geometry/cggg.bib}
}
@article{K18unique,
title = {{{There is a unique crossing-minimal rectilinear drawing of ${K}_{18}$}}},
author = {Bernardo M. {\'A}brego and Oswin Aichholzer and Silvia Fern{\'a}ndez-Merchant and Jes{\'u}s Lea{\~n}os and Gelasio Salazar},
journal = {Ars Mathematica Contemporanea},
year = {2024},
pages = {1--29},
volume = {24},
doi = {10.26493/1855-3974.2763.1e6},
originalfile = {/geometry/cggg.bib}
}
@article{Aichholzer_Brötzner_2024,
title = {{{Bicolored Order Types}}},
volume = {3},
url = {https://www.cgt-journal.org/index.php/cgt/article/view/46},
doi = {10.57717/cgt.v3i2.46},
abstract = {In their seminal work on Multidimensional Sorting, Goodman and Pollack introduced the so-called order type,
which for each ordered triple of a point set in the plane gives its orientation, clockwise or counterclockwise.
This information is sufficient to solve many problems from discrete geometry where properties of point sets do not depend on the exact coordinates
of the points but only on their relative positions. Goodman and Pollack showed that an efficient way to store an order type in a
matrix $\lambda$ of quadratic size (w.r.t.\ the number of points) is to count for every oriented line spanned by two points of
the set how many of the remaining points lie to the left of this line.
We generalize the concept of order types to bicolored point sets (every point has one of two colors).
The bicolored order type contains the orientation of each bicolored triple of points, while no information is stored for monochromatic triples.
Similar to the uncolored case, we store the number of blue points that are to the left of an oriented line spanned by two red points or by one red and one blue point in $\lambda_B$.
Analogously the number of red points is stored in $\lambda_R$. As a main result, we show that the equivalence of the information contained in the orientation of all bicolored point triples and the two matrices $\lambda_B$ and $\lambda_R$ also holds in the colored case.
This is remarkable, as in general the bicolored order type does not even contain sufficient information to determine all extreme points (points on the boundary of the convex hull of the point set).
We then show that the information of a bicolored order type is sufficient to determine whether the two color classes can be linearly separated and how one color class can be sorted around a point of the other color class.
Moreover, knowing the bicolored order type of a point set suffices to find bicolored plane perfect matchings or to compute the number of crossings of the complete bipartite graph drawn on a bicolored point set in quadratic time.},
number = {2},
journal = {Computing in Geometry and Topology},
author = {Aichholzer, Oswin and Br\"otzner, Anna},
year = {2024},
month = {Jan.},
pages = {3:1--3:17},
originalfile = {/geometry/cggg.bib}
}
@article{Aichholzer_Orthaber_Vogtenhuber_2024,
title = {{{Towards Crossing-Free Hamiltonian Cycles in Simple Drawings of Complete Graphs}}},
volume = {3},
url = {https://www.cgt-journal.org/index.php/cgt/article/view/47},
doi = {10.57717/cgt.v3i2.47},
abstract = {It is a longstanding conjecture that every simple drawing of a complete graph on $n\geq 3$ vertices contains a crossing-free Hamiltonian cycle. We strengthen this conjecture to \enquote{there exists a crossing-free Hamiltonian path between each pair of vertices} and show that this stronger conjecture holds for several classes of simple drawings, including strongly c-monotone drawings and cylindrical drawings. As a second main contribution, we give an overview on different classes of simple drawings and investigate inclusion relations between them up to weak isomorphism.},
number = {2},
journal = {Computing in Geometry and Topology},
author = {Aichholzer, Oswin and Orthaber, Joachim and Vogtenhuber, Birgit},
year = {2024},
month = {Jan.},
pages = {5:1--5:30},
originalfile = {/geometry/cggg.bib}
}
@article{afkpppsv-pmwc-24,
author = {Aichholzer, Oswin and Fabila-Monroy, Ruy and Kindermann, Philipp and Parada, Irene and Paul, Rosna and Perz, Daniel and Schnider, Patrick and Vogtenhuber, Birgit},
title = {{{Perfect Matchings with Crossings}}},
journal = {Algorithmica},
volume = {86},
pages = {697--716},
year = {2024},
abstract = {For sets of n points, n even, in general position in the plane, we consider straight-line drawings of perfect matchings on them. It is well known that such sets admit at least $$C_{n/2}$$different plane perfect matchings, where $$C_{n/2}$$is the n/2-th Catalan number. Generalizing this result we are interested in the number of drawings of perfect matchings which have k crossings. We show the following results. (1) For every $$k\le \frac{1}{64}n^2-\frac{35}{32}n\sqrt{n}+\frac{1225}{64}n$$, any set with n points, n sufficiently large, admits a perfect matching with exactly k crossings. (2) There exist sets of n points where every perfect matching has at most $$\frac{5}{72}n^2-\frac{n}{4}$$crossings. (3) The number of perfect matchings with at most k crossings is superexponential in n if k is superlinear in n. (4) Point sets in convex position minimize the number of perfect matchings with at most k crossings for $$k=0,1,2$$, and maximize the number of perfect matchings with $$\left( {\begin{array}{c}n/2\\ 2\end{array}}\right) $$crossings and with $${\left( {\begin{array}{c}n/2\\ 2\end{array}}\right) }\!-\!1$$crossings.},
doi = {10.1007/s00453-023-01147-7},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{avw-dtidcmg-23,
author = {Aichholzer, Oswin and Vogtenhuber, Birgit and Weinberger, Alexandra},
editor = {Bekos, Michael A. and Chimani, Markus},
title = {{"Different Types of Isomorphisms of Drawings of Complete Multipartite Graphs"}},
booktitle = {Graph Drawing and Network Visualization},
series = {Lecture Notes in Computer Science (LNCS)},
volume = {14466},
year = {2023},
publisher = {Springer Nature Switzerland},
address = {Cham},
pages = {34--50},
doi = {10.1007/978-3-031-49275-4_3},
eprint = {2308.10735},
archiveprefix = {arXiv},
abstract = {Simple drawings are drawings of graphs in which any two edges intersect at most once (either at a common endpoint or a proper crossing), and no edge intersects itself. We analyze several characteristics of simple drawings of complete multipartite graphs: which pairs of edges cross, in which order they cross, and the cyclic order around vertices and crossings, respectively. We consider all possible combinations of how two drawings can share some characteristics and determine which other characteristics they imply and which they do not imply. Our main results are that for simple drawings of complete multipartite graphs, the orders in which edges cross determine all other considered characteristics. Further, if all partition classes have at least three vertices, then the pairs of edges that cross determine the rotation system and the rotation around the crossings determine the extended rotation system. We also show that most other implications -- including the ones that hold for complete graphs -- do not hold for complete multipartite graphs. Using this analysis, we establish which types of isomorphisms are meaningful for simple drawings of complete multipartite graphs.},
isbn = {978-3-031-49275-4},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{afpsv-bpmc-23,
author = {Aichholzer, Oswin and Felsner, Stefan and Paul, Rosna and Scheucher, Manfred and Vogtenhuber, Birgit},
editor = {Bekos, Michael A. and Chimani, Markus},
title = {{"Bichromatic Perfect Matchings with Crossings"}},
booktitle = {Graph Drawing and Network Visualization},
series = {Lecture Notes in Computer Science (LNCS)},
volume = {14465},
year = {2023},
publisher = {Springer Nature Switzerland},
address = {Cham},
pages = {124--132},
doi = {10.1007/978-3-031-49272-3_9},
eprint = {2309.00546},
archiveprefix = {arXiv},
abstract = {We consider bichromatic point sets with n red and n blue points and study straight-line bichromatic perfect matchings on them. We show that every such point set in convex position admits a matching with at least {\$}{\$}{\backslash}frac{\{}3n^2{\}}{\{}8{\}}-{\backslash}frac{\{}n{\}}{\{}2{\}}+c{\$}{\$}3n28-n2+ccrossings, for some {\$}{\$} -{\backslash}frac{\{}1{\}}{\{}2{\}} {\backslash}le c {\backslash}le {\backslash}frac{\{}1{\}}{\{}8{\}}{\$}{\$}-12≤c≤18. This bound is tight since for any {\$}{\$}k> {\backslash}frac{\{}3n^2{\}}{\{}8{\}} -{\backslash}frac{\{}n{\}}{\{}2{\}}+{\backslash}frac{\{}1{\}}{\{}8{\}}{\$}{\$}k>3n28-n2+18there exist bichromatic point sets that do not admit any perfect matching with k crossings.},
isbn = {978-3-031-49272-3},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{abps-fom-24B,
title = {{{Flips in Odd Matchings}}},
author = {Aichholzer, Oswin and Brötzner, Anna and Perz, Daniel and Schnider, Patrick},
booktitle = {Proceedings of the 36th Canadian Conference on Computational Geometry (CCCG 2024)},
pages = {299--303},
year = {2024},
abstract = {Let $\mathcal{P}$ be a set of $n=2m+1$ points in the plane in general position.
We define the graph $GM_\mathcal{P}$ whose vertex set is the set of all plane matchings on $\mathcal{P}$ with exactly $m$~edges. Two vertices in $GM_\mathcal{P}$ are connected if the two corresponding matchings have $m-1$ edges in common. In this work we show that $GM_\mathcal{P}$ is connected.},
url = {https://cosc.brocku.ca/~rnishat/CCCG_2024_proceedings.pdf},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{abps-fom-24A,
title = {{{Flips in Odd Matchings}}},
author = {Aichholzer, Oswin and Brötzner, Anna and Perz, Daniel and Schnider, Patrick},
booktitle = {Proceedings of the 40th European Workshop on Computational Geometry (EuroCG 2024)},
pages = {59:1--59:6},
year = {2024},
abstract = {Let $P$ be a set of $n=2m+1$ points in the plane in general position.
We define the graph $GM_P$ whose vertex set is the set of all plane matchings on $P$ with exactly $m$~edges. Two vertices in $GM_P$ are connected if the two corresponding matchings have $m-1$ edges in common. In this work we show that $GM_P$ is connected.},
url = {https://eurocg2024.math.uoi.gr/data/uploads/paper_59.pdf},
originalfile = {/geometry/cggg.bib}
}
@inproceedings{-24,
title = {{{Connected matchings}}},
author = {Aichholzer, Oswin and Cabello, Sergio and M{\'e}sz{\'a}ros, Viola and Soukup, Jan},
booktitle = {Proceedings of the 40th European Workshop on Computational Geometry (EuroCG 2024)},
pages = {54:1--54:7},
year = {2024},
abstract = {We show that each set of $n \ge 2$ points in the plane in general position has a straight-line matching with at least $(5n+1)/27$ edges whose segments form a connected set, while for some point sets the best one can achieve is $\lceil \frac{n-1}{3}\rceil$.},
url = {https://eurocg2024.math.uoi.gr/data/uploads/paper_54.pdf},
originalfile = {/geometry/cggg.bib}
}
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