9758
2012
eng
article
1
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Augmenting the Connectivity of Planar and Geometric Graphs
In this paper we study connectivity augmentation problems. Given a connected graph G with some desirable property, we want to make G 2-vertex connected (or 2-edge connected) by adding edges such that the resulting graph keeps the property. The aim is to add as few edges as possible. The property that we consider is planarity, both in an abstract graph-theoretic and in a geometric setting, where vertices correspond to points in the plane and edges to straight-line segments.
We show that it is NP-hard to � nd a minimum-cardinality augmentation that makes a planar graph 2-edge connected. For making a planar graph 2-vertex connected this was known. We further show that both problems are hard in the geometric setting, even when restricted to trees. The problems remain hard for higher degrees of connectivity. On the other hand we give polynomial-time algorithms for the special case of convex geometric graphs.
We also study the following related problem. Given a planar (plane geometric) graph G, two vertices s and t of G, and an integer c, how many edges have to be added to G such that G is still planar (plane geometric) and contains c edge- (or vertex-) disjoint s{t paths? For the planar case we give a linear-time algorithm for c = 2. For the plane geometric case we give optimal worst-case bounds for c = 2; for c = 3 we characterize the cases that have a solution.
Journal of Graph Algorithms and Applications
10.7155/jgaa.00275
urn:nbn:de:bvb:20-opus-97587
In: Journal of Graph Algorithms and Applications (2012) 16: 2, 599-628, DOI: 10.755/jgaa.00275
Deutsches Urheberrecht
Alexander Wolff
Iganz Rutter
Datenverarbeitung; Informatik
open_access
Institut für Informatik
Universität Würzburg
https://opus.bibliothek.uni-wuerzburg.de/files/9758/Wolff_jgaa-00275.pdf
6699
2012
deu
article
1
2013-07-09
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Cover contact graphs
We study problems that arise in the context of covering certain geometric objects called seeds (e.g., points or disks) by a set of other geometric objects called cover (e.g., a set of disks or homothetic triangles). We insist that the interiors of the seeds and the cover elements are pairwise disjoint, respectively, but they can touch. We call the contact graph of a cover a cover contact graph (CCG). We are interested in three types of tasks, both in the general case and in the special case of seeds on a line: (a) deciding whether a given seed set has a connected CCG, (b) deciding whether a given graph has a realization as a CCG on a given seed set, and (c) bounding the sizes of certain classes of CCG’s. Concerning (a) we give efficient algorithms for the case that seeds are points and show that the problem becomes hard if seeds and covers are disks. Concerning (b) we show that this problem is hard even for point seeds and disk covers (given a fixed correspondence between graph vertices and seeds). Concerning (c) we obtain upper and lower bounds on the number of CCG’s for point seeds.
urn:nbn:de:bvb:20-opus-78845
7884
In: Journal of Computational Geometry (2012) 3(1), 102-131; http://jocg.org/index.php/jocg/article/view/66
Nieves Atienza
Natalia de Castro
Carmen Cortés
M. Ángeles Garrido
Clara I. Grima
Gregorio Hernández
Alberto Márquez
Auxiliadora Moreno-González
Martin Nöllenburg
José Ramón Portillo
Pedro Reyes
Jesús Valenzuela
Maria Trinidad Villar
Alexander Wolff
deu
swd
Informatik
Datenverarbeitung; Informatik
open_access
Institut für Informatik
Universität Würzburg
https://opus.bibliothek.uni-wuerzburg.de/files/6699/091_jocg_3_1.pdf
12462
2012
eng
309-332
62
article
1
2016-01-19
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Drawing (Complete) Binary Tanglegrams
A binary tanglegram is a drawing of a pair of rooted binary trees whose leaf sets are in one-to-one correspondence; matching leaves are connected by inter-tree edges. For applications, for example, in phylogenetics, it is essential that both trees are drawn without edge crossings and that the inter-tree edges have as few crossings as possible. It is known that finding a tanglegram with the minimum number of crossings is NP-hard and that the problem is fixed-parameter tractable with respect to that number.
We prove that under the Unique Games Conjecture there is no constant-factor approximation for binary trees. We show that the problem is NP-hard even if both trees are complete binary trees. For this case we give an O(n 3)-time 2-approximation and a new, simple fixed-parameter algorithm. We show that the maximization version of the dual problem for binary trees can be reduced to a version of MaxCut for which the algorithm of Goemans and Williamson yields a 0.878-approximation.
Algorithmica
10.1007/s00453-010-9456-3
urn:nbn:de:bvb:20-opus-124622
Algorithmica (2012) 62:309–332. doi:10.1007/s00453-010-9456-3
Kevin Buchin
Maike Buchin
Jaroslaw Byrka
Martin Nöllenburg
Yoshio Okamoto
Rodrigo I. Silveira
Alexander Wolff
eng
uncontrolled
NP-hardness
eng
uncontrolled
crossing minimization
eng
uncontrolled
binary tanglegram
eng
uncontrolled
approximation algorithm
eng
uncontrolled
fixed-parameter tractability
Datenverarbeitung; Informatik
open_access
Institut für Informatik
Universität Würzburg
https://opus.bibliothek.uni-wuerzburg.de/files/12462/art_10.1007_s00453-010-9456-3.pdf