Tight bound for the Erdős-Pósa property of tree minors

TL;DR

This paper resolves the Erdős–Pósa-type packing-versus-hitting question for minors of trees: for a tree $T$ on $t$ vertices, every graph $G$ and integer $k$ yields either $k$ vertex-disjoint subgraphs each containing a $T$ minor or a vertex set $X$ with $|X| \,\le\, t(k-1)$ that destroys all $T$ minors in $G-X$. The bound is tight and the authors extend the result to forests, obtaining $|X| \le tk - t'$ where $t'$ is the largest component size of the forest. A corollary provides a pathwidth-based packing/covering statement: for any $p$ and $k$, either $k$ disjoint subgraphs have pathwidth at least $p$ or a hitting set of size at most $2\cdot 3^{p+1}k reduces pathwidth below $p$. The proofs use a concise inductive framework and a Diestel-type pathwidth lemma to control the structure, avoiding heavy MSO machinery and yielding tight, linear-in-$k$ bounds in the tree/forest setting.

Abstract

Let $T$ be a tree on $t$ vertices. We prove that for every positive integer $k$ and every graph $G$, either $G$ contains $k$ pairwise vertex-disjoint subgraphs each having a $T$ minor, or there exists a set $X$ of at most $t(k-1)$ vertices of $G$ such that $G-X$ has no $T$ minor. The bound on the size of $X$ is best possible and improves on an earlier $f(t)k$ bound proved by Fiorini, Joret, and Wood (2013) with some fast growing function $f(t)$. Moreover, our proof is short and simple.

Figures (1)

• Figure 1: The set $Y$ and the graph $G_\ell$ whose boundary in $G$ is contained in $B_\ell$.

Theorems & Definitions (7)

• Theorem 1
• Theorem 2
• Corollary 3
• Lemma 4: D95
• proof : Proof of \ref{['thm:main-forest']}
• Lemma 5
• proof : Proof of \ref{['cor:pathwidth']}