Properties and Complexity of Fan-Planarity

TL;DR

This paper explores the properties, density bounds, and computational complexity of fan-planar graphs, extending existing results and analyzing their relationship with k-planarity, including NP-completeness of recognition.

Contribution

It provides tight bounds on fan-planar graph densities, extends previous results, and proves NP-completeness of fan-planarity recognition in variable embedding.

Findings

Fan-planar graphs have at most 5n-10 edges, tight for n ≥ 20.

Extended bounds for constrained fan-planar drawings.

Deciding fan-planarity is NP-complete.

Abstract

In a \emph{fan-planar drawing} of a graph an edge can cross only edges with a common end-vertex. Fan-planar drawings have been recently introduced by Kaufmann and Ueckerdt, who proved that every $n$-vertex fan-planar drawing has at most $5n-10$ edges, and that this bound is tight for $n \geq 20$. We extend their result, both from the combinatorial and the algorithmic point of view. We prove tight bounds on the density of constrained versions of fan-planar drawings and study the relationship between fan-planarity and $k$-planarity. Furthermore, we prove that deciding whether a graph admits a fan-planar drawing in the variable embedding setting is NP-complete.