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.
Bibliographical noteFunding Information:
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 (DIS-CONV). The second author was supported by the 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 the NKFIH under the grant SNN 116095. E-mail addresses: firstname.lastname@example.org (Eyal Ackerman) email@example.com (Balázs Keszegh) firstname.lastname@example.org (Mate Vizer)
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ASJC Scopus subject areas
- Theoretical Computer Science
- Computer Science (all)
- Computer Science Applications
- Geometry and Topology
- Computational Theory and Mathematics