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

Ackerman, Eyal

Keszegh, Balázs

Vizer, Mate

English

arXiv

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

Vizer, Máté

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