On the maximum crossing number

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

Research output: Contribution to journalArticlepeer-review

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, e.g., 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.

MSC Codes 68R10, 68U05

Original languageEnglish (US)
JournalUnknown Journal
StatePublished - May 15 2017

ASJC Scopus subject areas

  • General

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