Excluded Forest Minors and the Erd\H{o}s-P\'osa Property

TL;DR

This paper proves that for forest minors, the Erdős-Pósa function can be linear, improving previous bounds, while also establishing that such linear bounds are impossible for graphs with cycles.

Contribution

The paper demonstrates that the Erdős-Pósa function for forest minors is linear, providing the best possible bound for this case, and clarifies the limitations for graphs with cycles.

Findings

Linear Erdős-Pósa function for forest minors.

Exponential bounds for general graphs with cycles.

No linear bound possible if the minor contains a cycle.

Abstract

A classical result of Robertson and Seymour states that the set of graphs containing a fixed planar graph $H$ as a minor has the so-called Erd\H{o}s-P\'osa property; namely, there exists a function $f$ depending only on $H$ such that, for every graph $G$ and every positive integer $k$, the graph $G$ has $k$ vertex-disjoint subgraphs each containing $H$ as a minor, or there exists a subset $X$ of vertices of $G$ with $|X| \leq f(k)$ such that $G - X$ has no $H$-minor. While the best function $f$ currently known is exponential in $k$, a $O(k \log k)$ bound is known in the special case where $H$ is a forest. This is a consequence of a theorem of Bienstock, Robertson, Seymour, and Thomas on the pathwidth of graphs with an excluded forest-minor. In this paper we show that the function $f$ can be taken to be linear when $H$ is a forest. This is best possible in the sense that no linear bound is possible if $H$ has a cycle.