Gap-planar graphs

Abstract

We introduce the family of k-gap-planar graphs for k ≥ 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 kgap-planar graphs.

On fan-crossing and fan-crossing free graphs

“…Similar facts are known for 1-planar graphs [44], RAC graphs [30,34], 1-gapplanar graphs [13], and quasi-planar graphs [2,3,8]. For a survey see [31].…”

“…From the density or complete (bipartite) graphs and graph G from Lemma 4, we obtain non-fan-crossing graphs via additions of many short paths, so that we can conclude. Bae et al [13] state the relationship between 1-gap-planar and fan-planar (fan-crossing) graphs as an open problem and do not ask for the relationship between 1-gap-planar and fan-crossing free. We solve the latter.…”

“…Graph σ 3 (G) has n + 2m vertices. It has at most 5n + 10m − 10 edges if it were 1-gap-planar and O( √ k(n + m) for k ≥ 2 [13], so that a k-gap-planar drawing has at most that many crossings The complete graph K n has Ω(n 4 ) crossings [43]. The number of crossings of a graph is unchanged by subdivision.…”

A Survey on Graph Drawing Beyond Planarity

“…Density of fan-planar, k-fan-crossing-free, and k-gap graphs. Both fan-planar and 1-gap planar graphs on n vertices have at most 5n − 10 edges, which is a tight bound [154,31]. Recall that the same bound holds for 2-planar graphs.…”

Simple k-planar graphs are simple (k + 1)-quasiplanar

“…However, the relation between the hierarchies is not fully understood. For every k ≥ 3, there are infinitely many simple 3-quasiplanar graphs that are not simple k-planar [7]. Also, it is easy to see that, for k ≥ 1, every k-planar simple topological graph is (k + 2)quasiplanar.…”