Simplifying Non-Simple Fan-Planar Drawings

TL;DR

This paper proves that non-simple fan-planar graph drawings can be simplified to simple ones without increasing crossings, leading to bounds on edges and NP-hardness of recognition, resolving an open problem.

Contribution

It shows non-simple fan-planar drawings can be redrawn as simple ones without extra crossings, impacting graph density bounds and recognition complexity.

Findings

Non-simple fan-planar drawings can be simplified to simple drawings.

Graphs with such drawings have at most 6.5n edges.

Recognition of these graphs is NP-hard.

Abstract

A drawing of a graph is fan-planar if the edges intersecting a common edge $a$ share a vertex $A$ on the same side of $a$. More precisely, orienting $e$ arbitrarily and the other edges towards $A$ results in a consistent orientation of the crossings. So far, fan-planar drawings have only been considered in the context of simple drawings, where any two edges share at most one point, including endpoints. We show that every non-simple fan-planar drawing can be redrawn as a simple fan-planar drawing of the same graph while not introducing additional crossings. Combined with previous results on fan-planar drawings, this yields that $n$-vertex-graphs having such a drawing can have at most $6.5n$ edges and that the recognition of such graphs is NP-hard. We thereby answer an open problem posed by Kaufmann and Ueckerdt in 2014.