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Contraction-Bidimensionality of Geometric Intersection Graphs

Julien Baste, Dimitrios M. Thilikos, Marc Herbstritt
2018 International Symposium on Parameterized and Exact Computation  
This considerably extends the applicability of bidimensionality theory for several intersection graph classes of 2-dimensional geometrical objects.  ...  These results apply to a wide family of problems, namely problems that are contraction-bidimensional.  ...  Given a non-negative integer c, two graphs H and G, and a surjection σ : V (G) → V (H) we write H ≤ c σ G if 5:2 Contraction-Bidimensionality of Geometric Intersection Graphs for every x ∈ V (H), the  ... 
doi:10.4230/lipics.ipec.2017.5 dblp:conf/iwpec/BasteT17 fatcat:wfxlstjkhbdjhdvghnmlwka47a

Contraction Bidimensionality of Geometric Intersection Graphs [article]

Julien Baste, Dimitrios M. Thilikos
2022 arXiv   pre-print
This considerably extends the applicability of bidimensionality theory for contraction bidimensional problems.  ...  These results apply to a wide family of problems, namely problems that are contraction-bidimensional.  ...  As a result of this, the applicability of bidimensionality theory for minor-bidimensional problems has been extended to much wider families (not necessarily minor-closed) of graph classes of geometric  ... 
arXiv:2207.09751v1 fatcat:pley34pjujbhnl3yps52bofala

Bidimensionality of Geometric Intersection Graphs [chapter]

Alexander Grigoriev, Athanassios Koutsonas, Dimitrios M. Thilikos
2014 Lecture Notes in Computer Science  
We consider geometric intersection graphs G B where each body of the collection B is represented by a vertex, and two vertices of G B are adjacent if the intersection of the corresponding bodies is non-empty  ...  These combinatorial results vastly extend the applicability of all the meta-algorithmic results of the bidimensionality theory to geometrically defined graph classes.  ...  Acknowledgements The authors wish to thank all the anonymous reviewers for their valuable comments on previous versions of this paper.  ... 
doi:10.1007/978-3-319-04298-5_26 fatcat:dpzsfjp3wveejikqempkynkxsq

Bidimensionality of Geometric Intersection Graphs [article]

Alexander Grigoriev and Athanassios Koutsonas and Dimitrios M. Thilikos
2013 arXiv   pre-print
We consider geometric intersection graphs GB where each body of the collection B is represented by a vertex, and two vertices of GB are adjacent if the intersection of the corresponding bodies is non-empty  ...  These combinatorial results vastly extend the applicability of all the meta-algorithmic results of the bidimensionality theory to geometrically defined graph classes.  ...  Acknowledgements The authors wish to thank all the anonymous reviewers for their valuable comments on previous versions of this paper.  ... 
arXiv:1308.6166v1 fatcat:cbiokadsdnhahjrb7bhxqgoz5m

Bidimensionality and Parameterized Algorithms (Invited Talk)

Dimitrios M. Thilikos, Marc Herbstritt
2015 International Symposium on Parameterized and Exact Computation  
This theory applies to graph problems that are bidimensional in the sense that i) their solution value is not increasing when we take minors or contractions of the input graph and ii) their solution value  ...  We provide an exposition of the main results of the theory of bidimensionality in parameterized algorithm design.  ...  This result has been extended for more general families of geometric intersection graphs in [67] . Parameterized problems on graphs Parameterized problems.  ... 
doi:10.4230/lipics.ipec.2015.1 dblp:conf/iwpec/Thilikos15 fatcat:45a2txmf45bndj34diwdelb5na

Bidimensionality and Geometric Graphs [chapter]

Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh
2012 Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms  
We also show that our approach can not be extended in its full generality to more general classes of geometric graphs, such as intersection graphs of unit balls in R d , d ≥ 3.  ...  Our algorithms work directly on the input graph and do not require the geometric representations of the input graph.  ...  A graph H obtained by a sequence of edge-contractions is said to be a contraction of G. A graph H is a minor of a graph G if H is the contraction of some subgraph of G and we denote it by H ≤ m G.  ... 
doi:10.1137/1.9781611973099.124 dblp:conf/soda/FominLS12 fatcat:s7q42vkczjbp7drdztgo77ool4

Bidimensionality and Geometric Graphs [article]

Fedor V. Fomin and Daniel Lokshtanov and Saket Saurabh
2011 arXiv   pre-print
We proceed to show that our approach can not be extended in its full generality to more general classes of geometric graphs, such as intersection graphs of unit balls in R^d, d >= 3.  ...  Thus it is not necessary to compute the geometric representations of the input graph.  ...  A graph H obtained by a sequence of edge-contractions is said to be a contraction of G. A graph H is a minor of a graph G if H is the contraction of some subgraph of G and we denote it by H ≤ m G.  ... 
arXiv:1107.2221v2 fatcat:hc5q65lts5bl3jh5edwlx5l7w4

Optimality Program in Segment and String Graphs [chapter]

Édouard Bonnet, Paweł Rzążewski
2018 Lecture Notes in Computer Science  
Intrigued by an algorithm packing curves in 2 O(n 2/3 log n) by Fox and Pach [SODA'11], we investigate which problems have subexponential algorithms on the intersection graphs of curves (string graphs)  ...  or segments (segment intersection graphs) and which problems have no such algorithms under the ETH (Exponential Time Hypothesis).  ...  Subexponential algorithms on geometric intersection graphs. By a ply of a family of geometric objects we denote the maximum number of objects covering a single point.  ... 
doi:10.1007/978-3-030-00256-5_7 fatcat:63oa6burhndrbnpfoamjctqvsu

Computational Study on Bidimensionality Theory Based Algorithm for Longest Path Problem [chapter]

Chunhao Wang, Qian-Ping Gu
2011 Lecture Notes in Computer Science  
A parameter is bidimensional if it is either minor bidimensional or contraction bidimensional.  ...  The intuition behind bidimensionality theory is that some parameters do not increase under taking minors or contractions in a graph.  ... 
doi:10.1007/978-3-642-25591-5_38 fatcat:622ghx3ekrc6jlfa6rsxip3rte

Optimality Program in Segment and String Graphs

Édouard Bonnet, Paweł Rzążewski
2019 Algorithmica  
Intrigued by an algorithm packing curves in 2 O(n 2/3 log n) by Fox and Pach (SODA'11), we investigate which problems have subexponential algorithms on the intersection graphs of curves (string graphs)  ...  or segments (segment intersection graphs) and which problems have no such algorithms under the Exponential Time Hypothesis (ETH).  ...  Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution  ... 
doi:10.1007/s00453-019-00568-7 fatcat:y6rwhbhpdfhuzp4rhg65qx3z7e

Optimality program in segment and string graphs [article]

Édouard Bonnet, Paweł Rzążewski
2018 arXiv   pre-print
Intrigued by an algorithm packing curves in $2^{O(n^{2/3}\log n)}$ by Fox and Pach [SODA'11], we investigate which problems have subexponential algorithms on the intersection graphs of curves (string graphs  ...  ) or segments (segment intersection graphs) and which problems have no such algorithms under the ETH (Exponential Time Hypothesis).  ...  The second author is grateful to organizers of Homonolo 2016 and 2017, where part of the work was done, and also to Karolina Okrasa for inspiring discussions on the topic.  ... 
arXiv:1712.08907v3 fatcat:hrrbw622ova7hebjvirt6rmdwy

The Square Root Phenomenon in Planar Graphs [chapter]

Dániel Marx
2013 Lecture Notes in Computer Science  
, and contracting edges.Note: length of the longest path in H is at most the length of the longest path in G .  ...  elegant subexponential algorithms on planar graphs for parameterized problems such as Bidimensionality for k-Path Observation: If the treewidth of a planar graph G is at least 5 √ k ⇒ It has a √ k ×  ...  . ≤ relation between coordinates ⇐⇒ disks do not intersect. 50 Reduction to unit disks Every pair is represented by a unit disk in the plane. ≤ relation between coordinates ⇐⇒ disks do not intersect.  ... 
doi:10.1007/978-3-642-38756-2_1 fatcat:ldsbn2zrdje4tlferdy2z7cg5u

The Square Root Phenomenon in Planar Graphs [chapter]

Dániel Marx
2013 Lecture Notes in Computer Science  
, and contracting edges.Note: length of the longest path in H is at most the length of the longest path in G .  ...  elegant subexponential algorithms on planar graphs for parameterized problems such as Bidimensionality for k-Path Observation: If the treewidth of a planar graph G is at least 5 √ k ⇒ It has a √ k ×  ...  . ≤ relation between coordinates ⇐⇒ disks do not intersect. 50 Reduction to unit disks Every pair is represented by a unit disk in the plane. ≤ relation between coordinates ⇐⇒ disks do not intersect.  ... 
doi:10.1007/978-3-642-39212-2_4 fatcat:aoetw6pginbj3gkfymdj3f54fq

Computational topology of graphs on surfaces [article]

Éric Colin de Verdière
2017 arXiv   pre-print
curves and graphs on surfaces, and solving standard graph algorithm problems more efficiently in the case of surface-embedded graphs.  ...  Computational topology is an area that revisits topological problems from an algorithmic point of view, and develops topological tools for improved algorithms.  ...  The geometric intersection number of γ and δ is the minimum number of crossings between curves γ and δ isotopic to γ and δ.)  ... 
arXiv:1702.05358v2 fatcat:6vlahbtjnbaxhg4yy2zotfrsym

Finding, Hitting and Packing Cycles in Subexponential Time on Unit Disk Graphs [article]

Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, Meirav Zehavi
2017 arXiv   pre-print
Our algorithms are based on a new kind of tree decompositions of unit disk graphs where the separators can have size up to k^O(1) and there exists a solution that crosses every separator at most O(√(k)  ...  vertex set of size at most k, and (5) a set of k pairwise vertex-disjoint cycles.  ...  Then, the unit disk graph of D is the intersection graph of D.  ... 
arXiv:1704.07279v1 fatcat:57rulr7u3vcblod6tfw3uj2znm
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