Weakly and Strongly Fan-Planar Graphs

TL;DR

This paper investigates two variants of fan-planar graphs, demonstrating their differences in properties and establishing that both share the same maximum edge density despite their non-equivalence.

Contribution

It clarifies the distinction between weak and strong fan-planarity and proves that not all weakly fan-planar graphs are strongly fan-planar, while providing bounds on edge density.

Findings

Not all weakly fan-planar graphs are strongly fan-planar.

Both graph classes have the same upper bound on edge density.

Abstract

We study two notions of fan-planarity introduced by (Cheong et al., GD22), called weak and strong fan-planarity, which separate two non-equivalent definitions of fan-planarity in the literature. We prove that not every weakly fan-planar graph is strongly fan-planar, while the upper bound on the edge density is the same for both families.