Exact Crossing Number Parameterized by Vertex Cover

We prove that the exact crossing number of a graph can be efficiently computed for simple graphs having bounded vertex cover. In more precise words, Crossing Number is in FPT when parameterized by the vertex cover size. This is a notable advance since we know only very few nontrivial examples of gra...

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Published inGraph Drawing and Network Visualization Vol. 11904; pp. 307 - 319
Main Authors Hliněný, Petr, Sankaran, Abhisekh
Format Book Chapter
LanguageEnglish
Published Switzerland Springer International Publishing AG 2019
Springer International Publishing
SeriesLecture Notes in Computer Science
Subjects
Crossing number
Graph drawing
Parameterized complexity
Vertex cover
Online AccessGet full text
ISBN3030358011
9783030358013
ISSN0302-9743
1611-3349
DOI10.1007/978-3-030-35802-0_24

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Abstract We prove that the exact crossing number of a graph can be efficiently computed for simple graphs having bounded vertex cover. In more precise words, Crossing Number is in FPT when parameterized by the vertex cover size. This is a notable advance since we know only very few nontrivial examples of graph classes with unbounded and yet efficiently computable crossing number. Our result can be viewed as a strengthening of a previous result of Lokshtanov [arXiv, 2015] that Optimal Linear Arrangement is in FPT when parameterized by the vertex cover size, and we use a similar approach of reducing the problem to a tractable instance of Integer Quadratic Programming as in Lokshtanov’s paper.
AbstractList We prove that the exact crossing number of a graph can be efficiently computed for simple graphs having bounded vertex cover. In more precise words, Crossing Number is in FPT when parameterized by the vertex cover size. This is a notable advance since we know only very few nontrivial examples of graph classes with unbounded and yet efficiently computable crossing number. Our result can be viewed as a strengthening of a previous result of Lokshtanov [arXiv, 2015] that Optimal Linear Arrangement is in FPT when parameterized by the vertex cover size, and we use a similar approach of reducing the problem to a tractable instance of Integer Quadratic Programming as in Lokshtanov’s paper.
Author Sankaran, Abhisekh
Hliněný, Petr
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Notes P. Hliněný—Supported by the Czech Science Foundation, project no. 17-00837S.A. Sankaran—Supported by the Leverhulme Trust through a Research Project Grant on ‘Logical Fractals’.
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Snippet We prove that the exact crossing number of a graph can be efficiently computed for simple graphs having bounded vertex cover. In more precise words, Crossing...
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StartPage 307
SubjectTerms Crossing number
Graph drawing
Parameterized complexity
Vertex cover
Title Exact Crossing Number Parameterized by Vertex Cover
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