Erdős-Pósa property of rooted tree minors

2026-07-29

AI Summary

This work investigates whether rooted $T$-minors in a graph $G$, rooted at a vertex subset $S$, satisfy the Erdős–Pósa property. By generalizing Gallai’s theorem on $S$-paths to rooted minors of an arbitrary tree $T$, the authors establish that this property indeed holds: there exists a constant $c$ depending only on $T$ such that $G$ either contains $k$ vertex-disjoint rooted $T$-minors or admits a vertex hitting set of size at most $ck$ that intersects all such minors. This result improves the previously known $O(k^2)$ upper bound on the hitting set size to a linear $O(k)$ bound, which is tight in terms of $k$, thereby significantly advancing the theory in structural graph theory and combinatorial optimization.

Abstract

Fiorini, Joret, and Wood (2013) showed that tree minors satisfy the so-called Erdős-Pósa property with a linear bound: For every tree $T$ there exists a constant $c \geq 1$ such that, for every graph $G$ and integer $k\geq 0$, either $G$ contains $k$ vertex-disjoint subgraphs each containing a $T$-minor, or $G$ has a set $X$ of at most $c k$ vertices such that $G-X$ has no $T$-minor. In this paper, we prove that the same result remains true if, given a subset $S$ of vertices of $G$, one only considers $T$-minors of $G$ that are rooted in $S$. Here, a $T$-minor is rooted in $S$ if there is a minor-model of $T$ where each branch set contains a vertex from $S$. This result can be seen as a generalization of the classical $S$-Path Theorem of Gallai, which corresponds to the case $T=K_2$. The upper bound on the size of $X$ is best possible up to the value of the constant $c$, and improves on an earlier $O(k^2)$ bound due to Hodor, La, Micek, and Rambaud (2026).

Problem

• Erdős-Pósa property
• rooted tree minor
• vertex-disjoint subgraphs
• minor-model
• graph theory

Innovation

• Erdős–Pósa property
• rooted tree minor
• linear bound
• S-path theorem
• graph minor