Journal of Graph Algorithms and Applications
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Special Issue on Graph Drawing Beyond Planarity
On the size of planarly connected crossing graphs
Eyal Ackerman, Balázs Keszegh, and Mate Vizer
Vol. 22, no. 1, pp. 11-22, 2018. Regular paper.
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.
Submitted: April 2017.
Reviewed: June 2017.
Revised: July 2017.
Accepted: July 2017.
Final: July 2017.
Published: January 2018.