Pair crossing number, cutwidth, and good drawings on arbitrary point sets

TL;DR

This paper establishes new bounds relating crossing numbers, pair crossing numbers, and cutwidth of graphs, and demonstrates how to draw graphs with controlled crossings on arbitrary point sets, advancing geometric graph theory.

Contribution

It improves bounds on crossing numbers in terms of pair crossing numbers and provides a method for straight-line graph drawings with few crossings on arbitrary point sets.

Findings

Bound: cr(G)=O(pcr(G)^{3/2}) improves previous results.

Bound: bw(G)=O(√(pcr(G)+∑d_k^2)) answers a question by Pach and Toth.

Graphs can be drawn with O(log n) crossings on arbitrary point sets.

Abstract

Determining whether there exists a graph such that its crossing number and pair crossing number are distinct is an important open problem in geometric graph theory. We show that $\textit{cr}(G)=O(\mathop{\mathrm{pcr}}(G)^{3/2})$ for every graph $G$, this improves the previous best bound by a logarithmic factor. Answering a question of Pach and T\'oth, we prove that the bisection width (and, in fact, the cutwidth as well) of a graph $G$ with degree sequence $d_1,d_2,\dots,d_n$ satisfies $\mathop{\mathrm{bw}}(G)=O\big(\sqrt{\mathop{\mathrm{pcr}}(G)+\sum_{k=1}^n d_k^2}\big)$. Then we show that there is a constant $C\geq 1$ such that the following holds: For any graph $G$ of order $n$ and any set $S$ of at least $n^C$ points in general position on the plane, $G$ admits a straight-line drawing which maps the vertices to points of $S$ and has no more than $O\left(\log n\cdot\left(\mathop{\mathrm{pcr}}(G)+\sum_{k=1}^n d_k^2\right)\right)$ crossings. Our proofs rely on a modified version of a separator theorem for string graphs by Lee, which might be of independent interest.