Bounds on treewidth via excluding disjoint unions of cycles

TL;DR

This paper establishes tighter bounds on the treewidth of graphs excluding disjoint unions of cycles as minors, improving previous bounds from polynomial to near-linear in the size of the cycles.

Contribution

It provides a nearly optimal bound of O(|V(H)| log^2 |V(H)|) for the treewidth of graphs excluding disjoint unions of cycles as minors, advancing understanding in graph minor theory.

Findings

Bound of O(|V(H)| log^2 |V(H)|) for treewidth when excluding disjoint unions of cycles

Improved from previous polynomial bounds for planar graphs

Near-optimal bound close to theoretical lower limit

Abstract

One of the fundamental results in graph minor theory is that for every planar graph~$H$, there is a minimum integer~$f(H)$ such that graphs with no minor isomorphic to~$H$ have treewidth at most~$f(H)$. The best known bound for an arbitrary planar $H$ is ${O(|V(H)|^9\operatorname{poly~log} |V(H)|)}$. We show that if $H$ is the disjoint union of cycles, then $f(H)$ is $O(|V(H)|\log^2 |V(H)|)$, which is a $\log|V(H)|$ factor away being optimal.