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.
['Eyal Ackerman', 'Balázs Keszegh', 'Máté Vizer']
On the size of planarly connected crossing graphs