Abstract
In a fan-planar drawing of a graph an edge can cross only edges with a common end-vertex. Fan-planar drawings have been recently introduced by Kaufmann and Ueckerdt, who proved that every n-vertex fan-planar drawing has at most 5n − 10 edges, and that this bound is tight for n ≥ 20. We extend their result from both the combinatorial and the algorithmic point of view. We prove tight bounds on the density of constrained versions of fan-planar drawings and study the relationship between fan-planarity and k-planarity. Also, we prove that testing fan-planarity in the variable embedding setting is NP-complete.
Research supported in part by the MIUR project AMANDA “Algorithmics for MAssive and Networked DAta”, prot. 2012C4E3KT_001. This work started at the Bertinoro Workshop on Graph Drawing 2014. We thank Michael Kaufmann and Torsten Ueckerdt for suggesting the study of fan-planar graphs during the workshop. We also thank all the participants of the workshop for the useful discussions on this topic.
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Binucci, C., Di Giacomo, E., Didimo, W., Montecchiani, F., Patrignani, M., Tollis, I.G. (2014). Fan-Planar Graphs: Combinatorial Properties and Complexity Results. In: Duncan, C., Symvonis, A. (eds) Graph Drawing. GD 2014. Lecture Notes in Computer Science, vol 8871. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-45803-7_16
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