On the maximum crossing number

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

Research output: Contribution to journalArticle

2 Scopus citations

Abstract

Research about crossings is typically about minimization. In this pa- per, 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 admits a non-convex drawing with more crossings than any convex drawing. 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 approx- imation even for the unweighted geometric case. We also prove that the unweighted topological case is NP-hard.

Original languageEnglish (US)
Pages (from-to)67-87
Number of pages21
JournalJournal of Graph Algorithms and Applications
Volume22
Issue number1
DOIs
StatePublished - Jan 1 2018

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)
  • Computer Science Applications
  • Geometry and Topology
  • Computational Theory and Mathematics

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    Chimani, M., Felsner, S., Kobourov, S., Ueckerdt, T., Valtr, P., & Wolff, A. (2018). On the maximum crossing number. Journal of Graph Algorithms and Applications, 22(1), 67-87. https://doi.org/10.7155/jgaa.00458