On Optimal Beyond-Planar Graphs

TL;DR

This paper investigates the properties and ranges of optimal beyond-planar graphs, which are graphs drawable with specific crossing restrictions, and demonstrates their relation to topological minors.

Contribution

It computes the ranges for optimal graphs across various beyond-planar types, establishes their combinatorial properties, and proves every graph is a topological minor of an optimal graph.

Findings

Determined the density ranges for optimal graphs in several beyond-planar classes.

Established combinatorial properties of these optimal graphs.

Proved that every graph is a topological minor of an optimal beyond-planar graph.

Abstract

A graph is beyond-planar if it can be drawn in the plane with a specific restriction on crossings. Several types of beyond-planar graphs have been investigated, such as k-planar if every edge is crossed at most k times and RAC if edges can cross only at a right angle in a straight-line drawing. A graph is optimal if the number of edges coincides with the density for its type. Optimal graphs are special and are known only for some types of beyond-planar graphs, including 1-planar, 2-planar, and RAC graphs. For all types of beyond-planar graphs for which optimal graphs are known, we compute the range for optimal graphs, establish combinatorial properties, and show that every graph is a topological minor of an optimal graph. Note that the minor property is well-known for general beyond-planar graphs.