Two-Planar Graphs Are Quasiplanar

TL;DR

This paper proves that every 2-planar graph can be drawn in a way that avoids three edges crossing pairwise, establishing a key relationship between 2-planar and quasiplanar graphs.

Contribution

It demonstrates that 2-planar graphs are quasiplanar and that quasiplanarity can be represented by simple topological drawings, advancing understanding of graph crossing properties.

Findings

Every 2-planar graph is quasiplanar.

Quasiplanarity can be realized with simple topological drawings.

Edges in such drawings cross at most once, and adjacent edges do not cross.

Abstract

It is shown that every 2-planar graph is quasiplanar, that is, if a simple graph admits a drawing in the plane such that every edge is crossed at most twice, then it also admits a drawing in which no three edges pairwise cross. We further show that quasiplanarity is witnessed by a simple topological drawing, that is, any two edges cross at most once and adjacent edges do not cross.