Geodesic
On the size of planarly connected crossing graphs
Journal of Graph Algorithms and Applications, Special Issue on Graph Drawing Beyond Planarity , Tome 22 (2018) no. 1, pp. 11-22.

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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. 
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     author = {Eyal Ackerman and Bal\'azs Keszegh and Mate Vizer},
     title = {On the size of planarly connected crossing graphs},
     journal = {Journal of Graph Algorithms and Applications},
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Eyal Ackerman; Balázs Keszegh; Mate Vizer. On the size of planarly connected crossing graphs. Journal of Graph Algorithms and Applications, 
							Special Issue on Graph Drawing Beyond Planarity
					, Tome 22 (2018) no. 1, pp. 11-22. doi : 10.7155/jgaa.00453. http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00453/

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