Journal of Graph Algorithms and Applications
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Special Issue on Graph Drawing Beyond Planarity
On the Maximum Crossing Number
Markus Chimani, Stefan Felsner, Stephen Kobourov, Torsten Ueckerdt, Pavel Valtr, and Alexander Wolff
Vol. 22, no. 1, pp. 67-87, 2018. Regular paper.
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 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 approximation even for the unweighted geometric case. We also prove that the unweighted topological case is NP-hard.
Submitted: May 2017.
Reviewed: August 2017.
Revised: October 2017.
Reviewed: December 2017.
Revised: December 2017.
Accepted: December 2017.
Final: December 2017.
Published: January 2018.