Convex-Geometric k-Planar Graphs Are Convex-Geometric \((k+1)\) -Quasiplanar

A drawing of a graph is k-planar if every edge has at most k crossings with other edges of the graph and it is k-quasiplanar if it has no set of k pairwise crossing edges. We say that a graph drawing is simple if two edges intersect at most once. In 2020, Angelini et al. proved that all simple k-planar graphs are simple \((k+1)\) -quasiplanar, which was the first non-trivial relationship between these two classes. We say that a graph drawing is convex-geometric if its vertices are drawn as points on a circle and its edges are drawn as straight line segments between them. In this paper we prove that, for \(k\ge 2\) , every convex-geometric k-planar graph is convex-geometric \((k+1)\) -quasiplanar.