TY - GEN

T1 - On the maximum crossing number

AU - Chimani, Markus

AU - Felsner, Stefan

AU - Kobourov, Stephen G

AU - Ueckerdt, Torsten

AU - Valtr, Pavel

AU - Wolff, Alexander

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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.

AB - 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.

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U2 - 10.1007/978-3-319-78825-8_6

DO - 10.1007/978-3-319-78825-8_6

M3 - Conference contribution

AN - SCOPUS:85046005203

SN - 9783319788241

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 61

EP - 74

BT - Combinatorial Algorithms - 28th International Workshop, IWOCA 2017, Revised Selected Papers

PB - Springer-Verlag

T2 - 28th International Workshop on Combinational Algorithms, IWOCA 2017

Y2 - 17 July 2017 through 21 July 2017

ER -