Gap-planar Graphs

TL;DR

This paper introduces $k$-gap-planar graphs, a new class of graphs with bounded crossings per edge, explores their density, relationships to other graph classes, and the complexity of recognizing them.

Contribution

It defines $k$-gap-planar graphs, analyzes their properties, and studies recognition complexity, advancing understanding of beyond-planar graph classes.

Findings

Maximum density bounds for $k$-gap-planar graphs

Characterization of $k$-gap-planar complete graphs

NP-completeness of recognition problem

Abstract

We introduce the family of $k$-gap-planar graphs for $k \geq 0$, i.e., graphs that have a drawing in which each crossing is assigned to one of the two involved edges and each edge is assigned at most $k$ of its crossings. This definition is motivated by applications in edge casing, as a $k$-gap-planar graph can be drawn crossing-free after introducing at most $k$ local gaps per edge. We present results on the maximum density of $k$-gap-planar graphs, their relationship to other classes of beyond-planar graphs, characterization of $k$-gap-planar complete graphs, and the computational complexity of recognizing $k$-gap-planar graphs.