On the size of planarly connected crossing graphs

TL;DR

This paper proves that graphs with certain planarly connected crossing drawings have a linear number of edges, contributing to the understanding of quasi-planar and related graph classes.

Contribution

It establishes an upper bound of O(n) edges for graphs with planarly connected crossing drawings, linking these to quasi-planar and 1-planar graphs.

Findings

Graphs with planarly connected crossing drawings have at most O(n) edges.

The result connects these graphs to well-studied classes like quasi-planar and 1-planar graphs.

Provides a new structural insight into the edge complexity of such graphs.

Abstract

We prove that if an $n$-vertex graph $G$ can be drawn in the plane such that each pair of crossing edges is independent and there is a crossing-free edge that connects their endpoints, then $G$ has $O(n)$ edges. Graphs that admit such drawings are related to quasi-planar graphs and to maximal $1$-planar and fan-planar graphs.