A Note on IC-Planar Graphs

TL;DR

This paper investigates IC-planar graphs, a class of graphs with specific crossing restrictions, establishing their maximum density and providing insights into their structural properties.

Contribution

It proves the existence of infinitely many maximal IC-planar graphs with a tight lower bound on their density, advancing understanding of their structural limits.

Findings

Existence of infinitely many maximal IC-planar graphs with n vertices and 3n-5 edges

Established a tight lower bound on the density of IC-planar graphs

Characterized the structural properties of IC-planar graphs

Abstract

A graph is IC-planar if it admits a drawing in the plane with at most one crossing per edge and such that two pairs of crossing edges share no common end vertex. IC-planarity specializes both NIC-planarity, which allows a pair of crossing edges to share at most one vertex, and 1-planarity, where each edge may be crossed at most once. We show that there are infinitely maximal IC-planar graphs with n vertices and 3n-5 edges and thereby prove a tight lower bound on the density of this class of graphs.