On the Size of Planarly Connected Crossing Graphs
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.
KeywordsPlanar graphs Crossing edges Crossing-free edge Fan-planar graphs 1-planar graphs
We thank Géza Tóth for his permission to include his construction for a lower bound on the size of a PCC graph in this paper. We also thank an anonymous referee for pointing out an error in an earlier version of this paper.
Most of this work was done during a visit of the first author to the Rényi Institute that was partially supported by the National Research, Development and Innovation Office – NKFIH under the grant PD 108406 and by the ERC Advanced Research Grant no. 267165 (DISCONV). The second author was supported by the National Research, Development and Innovation Office – NKFIH under the grant PD 108406 and K 116769 and by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. The third author was supported by Development and Innovation Office – NKFIH under the grant SNN 116095.
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