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

TL;DR

The paper studies the function $f(H)$ for $H$ the disjoint union of $k$ cycles, aiming to bound treewidth of graphs excluding $H$ as a minor. The main result proves $f(H) \le 6|V(H)| + 10 k log k + 10 k log log k + 40 k$, which is tight up to constants relative to the lower bound $\Omega(|V(H)| + k log k)$. The approach avoids bramble-based proofs and combines a refined Erdős–Pósa analysis for long cycles (via MNSW17) with Birmele’s bound on treeswidth for graphs with no long cycles, implemented through an inductive scheme using $g(h,k)$. Consequently, the work advances understanding of minor-closed families built from unions of cycles and provides explicit, near-optimal bounds on $f(H)$.

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.

Theorems & Definitions (5)

• Theorem 1
• Theorem 2: MNSW17
• Theorem 3: MNSW17, implicit
• Theorem 4: B03
• proof : Proof of \ref{['thm:main']}