On the Maximum Crossing Number Markus Chimani, Stefan Felsner, Stephen Kobourov, Torsten Ueckerdt, Pavel Valtr, and Alexander Wolff 1 Universitat¨ Osnabruck,¨
[email protected] 2 Technische Universitat¨ Berlin,
[email protected] 3 University of Arizona,
[email protected] 4 Karlsruhe Institute of Technology,
[email protected] 5 Charles University,
[email protected] 6 Universitat¨ Wurzburg,¨ orcid.org/0000-0001-5872-718X 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. [1] 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. [2] 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. 1 Introduction While traditionally in graph drawing one wants to minimize the number of edge cross- ings, we are interested in the opposite problem. Specifically, given a graph G, what is the maximum number of edge crossings possible, and what do embeddings7 of G that attain this maximum look like? Such questions have first been asked as early as in the 19th century [3,22].