Structure of Graphs with Locally Restricted Crossings

TL;DR

This paper establishes tight bounds relating the size, treewidth, and local crossing number of graphs embedded on surfaces, with implications for k-planar and map graphs, and explores embeddings with controlled crossings.

Contribution

It provides new tight bounds on treewidth and layered treewidth for graphs with bounded crossings per edge on surfaces, improving previous results and extending to map graphs.

Findings

Treewidth of graphs on surfaces with bounded crossings per edge is $O(\sqrt{(g+1)(k+1)n})$

Layered treewidth bounds are $O((g+1)k)$, tight up to constants

Graphs can be embedded with $O((m/(g+1))\log^2 g)$ crossings per edge, tight to polylogarithmic factors

Abstract

We consider relations between the size, treewidth, and local crossing number (maximum number of crossings per edge) of graphs embedded on topological surfaces. We show that an $n$-vertex graph embedded on a surface of genus $g$ with at most $k$ crossings per edge has treewidth $O(\sqrt{(g+1)(k+1)n})$ and layered treewidth $O((g+1)k)$, and that these bounds are tight up to a constant factor. As a special case, the $k$-planar graphs with $n$ vertices have treewidth $O(\sqrt{(k+1)n})$ and layered treewidth $O(k+1)$, which are tight bounds that improve a previously known $O((k+1)^{3/4}n^{1/2})$ treewidth bound. Analogous results are proved for map graphs defined with respect to any surface. Finally, we show that for $g<m$, every $m$-edge graph can be embedded on a surface of genus~$g$ with $O((m/(g+1))\log^2 g)$ crossings per edge, which is tight to a polylogarithmic factor.