Note on the treewidth of graphs excluding a disjoint union of cycles as a minor

TL;DR

This paper determines the bounds on the treewidth of planar graphs that exclude a disjoint union of cycles as a minor, establishing tight bounds related to the number of cycles and vertices.

Contribution

The paper provides a tight bound on the treewidth of graphs excluding a disjoint union of cycles as minors, extending understanding of graph minors and treewidth.

Findings

f(H)=O(|V(H)| + k log k) for disjoint union of k cycles

Bound is proven to be tight

Advances the theory of graph minors and treewidth

Abstract

For a planar graph $H$, let $f(H)$ denote the minimum integer such that all graphs excluding $H$ as a minor have treewidth at most $f(H)$. We show that if $H$ is a disjoint union of $k$ cycles then $f(H)=O(|V(H)| + k \log k)$, which is best possible.