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On the size of planarly connected crossing graphs
Published 8 Sep 2015 in math.CO and cs.CG | (1509.02475v3)
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.
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