On the size of planarly connected crossing graphs

Eyal Ackerman; Balázs Keszegh; Mate Vizer

On the size of planarly connected crossing graphs

Combinatorics 2016-08-31 v3 Computational Geometry

Authors: Eyal Ackerman , Balázs Keszegh , Mate Vizer

Abstract

We prove that if an $n$ -vertex graph $G$ can be drawn in the plane such that each pair of crossing edges is independent and there is a crossing-free edge that connects their endpoints, then $G$ has $O(n)$ edges. Graphs that admit such drawings are related to quasi-planar graphs and to maximal $1$ -planar and fan-planar graphs.

Keywords

crossing graphs planar connectivity quasi-planar graphs maximal 1-planar graphs fan-planar graphs extremal graph theory graph drawing

Cite

@article{arxiv.1509.02475,
  title  = {On the size of planarly connected crossing graphs},
  author = {Eyal Ackerman and Balázs Keszegh and Mate Vizer},
  journal= {arXiv preprint arXiv:1509.02475},
  year   = {2016}
}

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Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016)

On the size of planarly connected crossing graphs

Abstract

Keywords

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