On the maximum crossing number

Markus Chimani, Stefan Felsner, Stephen G Kobourov, Torsten Ueckerdt, Pavel Valtr, Alexander Wolff

Research output: Chapter in Book/Report/Conference proceedingConference contribution

2 Scopus citations

Abstract

Research about crossings is typically about minimization. In this paper, we consider maximizing the number of crossings over all possible ways to draw a given graph in the plane. Alpert et al. [Electron. J. Combin., 2009] conjectured that any graph has a convex straight-line drawing, that is, a drawing with vertices in convex position, that maximizes the number of edge crossings. We disprove this conjecture by constructing a planar graph on twelve vertices that allows a non-convex drawing with more crossings than any convex one. Bald et al. [Proc. COCOON, 2016] showed that it is NP-hard to compute the maximum number of crossings of a geometric graph and that the weighted geometric case is NP-hard to approximate. We strengthen these results by showing hardness of approximation even for the unweighted geometric case and prove that the unweighted topological case is NP-hard.

Original languageEnglish (US)
Title of host publicationCombinatorial Algorithms - 28th International Workshop, IWOCA 2017, Revised Selected Papers
PublisherSpringer-Verlag
Pages61-74
Number of pages14
ISBN (Print)9783319788241
DOIs
StatePublished - Jan 1 2018
Event28th International Workshop on Combinational Algorithms, IWOCA 2017 - Newcastle, NSW, Australia
Duration: Jul 17 2017Jul 21 2017

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume10765 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other28th International Workshop on Combinational Algorithms, IWOCA 2017
CountryAustralia
CityNewcastle, NSW
Period7/17/177/21/17

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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