@inproceedings{7021efd97f8d4aeaa80fce34ab562f73,

title = "On the maximum crossing number",

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.",

author = "Markus Chimani and Stefan Felsner and Stephen Kobourov and Torsten Ueckerdt and Pavel Valtr and Alexander Wolff",

note = "Publisher Copyright: {\textcopyright} Springer International Publishing AG, part of Springer Nature 2018.; 28th International Workshop on Combinational Algorithms, IWOCA 2017 ; Conference date: 17-07-2017 Through 21-07-2017",

year = "2018",

doi = "10.1007/978-3-319-78825-8_6",

language = "English (US)",

isbn = "9783319788241",

series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",

publisher = "Springer-Verlag",

pages = "61--74",

editor = "Smyth, {William F.} and Ljiljana Brankovic and Joe Ryan",

booktitle = "Combinatorial Algorithms - 28th International Workshop, IWOCA 2017, Revised Selected Papers",

}