Bounds on treewidth via excluding disjoint unions of cycles
Meike Hatzel, Chun-Hung Liu, Bruce Reed, Sebastian Wiederrecht
TL;DR
The paper studies f(H) for graphs H that are disjoint unions of cycles, seeking tighter bounds on the treewidth of graphs excluding H as a minor. It introduces a refined bound f(H) <= c|V(H)| log (r+1) + cr log r log ell for H consisting of r cycles of length at most ell, thereby achieving f(H) = O(|V(H)| log^2 |V(H)|) in general and improving prior O(|V(H)|^2 log |V(H)|) results. The authors deploy bramble theory, Erdős–Pósa, and path-partition/ linkage techniques to perform a detailed inductive proof on r, handling cases based on the longest cycle length and bramble interactions to either reduce H or construct the required minor. This advances understanding of how cycle-structure in H dictates treewidth bounds and yields near-optimal results in the regime of unions of cycles, with implications for algorithmic applications via fixed-parameter tractability tied to treewidth.
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.