# Crossing Numbers of Beyond-Planar Graphs

## TL;DR

This paper investigates the crossing numbers of beyond-planar graphs like 1-planar, quasi-planar, and fan-planar, showing that restricted drawings can have significantly more crossings than unrestricted minimal drawings.

## Contribution

It establishes lower bounds on crossings in restricted graph drawings and compares these to the minimal crossing numbers without restrictions.

## Key findings

- Restricted drawings can have linear crossings in the number of vertices.
- Unrestricted crossing-minimal drawings can have constant crossings.
- Different beyond-planar graph classes exhibit distinct crossing number behaviors.

## Abstract

We study the 1-planar, quasi-planar, and fan-planar crossing number in comparison to the (unrestricted) crossing number of graphs. We prove that there are $n$-vertex 1-planar (quasi-planar, fan-planar) graphs such that any 1-planar (quasi-planar, fan-planar) drawing has $\Omega(n)$ crossings, while $O(1)$ crossings suffice in a crossing-minimal drawing without restrictions on local edge crossing patterns.