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

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.