Geodesic
On the Maximum Crossing Number
Journal of Graph Algorithms and Applications, Special Issue on Graph Drawing Beyond Planarity , Tome 22 (2018) no. 1, pp. 67-87.

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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.
DOI : 10.7155/jgaa.00458
Keywords: graph drawing, maximum crossing number, maximum rectilinear crossing number, maximum convex crossing number, NP-hard, APX-hard 
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Markus Chimani; Stefan Felsner; Stephen Kobourov; Torsten Ueckerdt; Pavel Valtr; Alexander Wolff. On the Maximum Crossing Number. Journal of Graph Algorithms and Applications, 
							Special Issue on Graph Drawing Beyond Planarity
					, Tome 22 (2018) no. 1, pp. 67-87. doi : 10.7155/jgaa.00458. http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00458/

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