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Contraction-Bidimensionality of Geometric Intersection Graphs

2018
*
International Symposium on Parameterized and Exact Computation
*

This considerably extends the applicability

doi:10.4230/lipics.ipec.2017.5
dblp:conf/iwpec/BasteT17
fatcat:wfxlstjkhbdjhdvghnmlwka47a
*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 ...##
###
Contraction Bidimensionality of Geometric Intersection Graphs
[article]

2022
*
arXiv
*
pre-print

This considerably extends the applicability

arXiv:2207.09751v1
fatcat:pley34pjujbhnl3yps52bofala
*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*...##
###
Bidimensionality of Geometric Intersection Graphs
[chapter]

2014
*
Lecture Notes in Computer Science
*

We consider

doi:10.1007/978-3-319-04298-5_26
fatcat:dpzsfjp3wveejikqempkynkxsq
*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. ...##
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Bidimensionality of Geometric Intersection Graphs
[article]

2013
*
arXiv
*
pre-print

We consider

arXiv:1308.6166v1
fatcat:cbiokadsdnhahjrb7bhxqgoz5m
*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. ...##
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Bidimensionality and Parameterized Algorithms (Invited Talk)

2015
*
International Symposium on Parameterized and Exact Computation
*

This theory applies to

doi:10.4230/lipics.ipec.2015.1
dblp:conf/iwpec/Thilikos15
fatcat:45a2txmf45bndj34diwdelb5na
*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. ...##
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Bidimensionality and Geometric Graphs
[chapter]

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

doi:10.1137/1.9781611973099.124
dblp:conf/soda/FominLS12
fatcat:s7q42vkczjbp7drdztgo77ool4
*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. ...##
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Bidimensionality and Geometric Graphs
[article]

2011
*
arXiv
*
pre-print

We proceed to show that our approach can not be extended in its full generality to more general classes

arXiv:1107.2221v2
fatcat:hc5q65lts5bl3jh5edwlx5l7w4
*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. ...##
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Optimality Program in Segment and String Graphs
[chapter]

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

doi:10.1007/978-3-030-00256-5_7
fatcat:63oa6burhndrbnpfoamjctqvsu
*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. ...##
###
Computational Study on Bidimensionality Theory Based Algorithm for Longest Path Problem
[chapter]

2011
*
Lecture Notes in Computer Science
*

A parameter is

doi:10.1007/978-3-642-25591-5_38
fatcat:622ghx3ekrc6jlfa6rsxip3rte
*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*. ...##
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Optimality Program in Segment and String Graphs

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

doi:10.1007/s00453-019-00568-7
fatcat:y6rwhbhpdfhuzp4rhg65qx3z7e
*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 ...##
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Optimality program in segment and string graphs
[article]

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

arXiv:1712.08907v3
fatcat:hrrbw622ova7hebjvirt6rmdwy
*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. ...##
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The Square Root Phenomenon in Planar Graphs
[chapter]

2013
*
Lecture Notes in Computer Science
*

, and

doi:10.1007/978-3-642-38756-2_1
fatcat:ldsbn2zrdje4tlferdy2z7cg5u
*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*. ...##
###
The Square Root Phenomenon in Planar Graphs
[chapter]

2013
*
Lecture Notes in Computer Science
*

, and

doi:10.1007/978-3-642-39212-2_4
fatcat:aoetw6pginbj3gkfymdj3f54fq
*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*. ...##
###
Computational topology of graphs on surfaces
[article]

2017
*
arXiv
*
pre-print

curves and

arXiv:1702.05358v2
fatcat:6vlahbtjnbaxhg4yy2zotfrsym
*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.) ...##
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Finding, Hitting and Packing Cycles in Subexponential Time on Unit Disk Graphs
[article]

2017
*
arXiv
*
pre-print

Our algorithms are based on a new kind

arXiv:1704.07279v1
fatcat:57rulr7u3vcblod6tfw3uj2znm
*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. ...
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