Polynomial Gap Extensions of the Erd\H{o}s-P\'osa Theorem

TL;DR

This paper extends the Erdős-Pósa theorem to graphs with pathwidth at most 2, providing bounds on the function relating disjoint subgraphs and hitting sets, with improved bounds for specific graph classes.

Contribution

It proves the Erdős-Pósa extension for graphs of pathwidth at most 2 and refines bounds for particular cases like $K_{2,r}$, improving understanding of graph minors and treewidth.

Findings

Established Erdős-Pósa extension for pathwidth ≤ 2 graphs.

Derived bounds on the function $f_H(k)$ for these graphs.

Improved bounds for the special case $H=K_{2,r}$.

Abstract

Given a graph $H$, we denote by ${\cal M}(H)$ all graphs that can be contracted to $H$. The following extension of the Erd\H{o}s-P\'osa Theorem holds: for every $h$-vertex planar graph $H$, there exists a function $f_{H}$ such that every graph $G$, either contains $k$ disjoint copies of graphs in ${\cal M}(H)$, or contains a set of $f_{H}(k)$ vertices meeting every subgraph of $G$ that belongs in ${\cal M}(H)$. In this paper we prove that this is the case for every graph $H$ of pathwidth at most 2 and, in particular, that $f_{H}(k) = 2^{O(h^2)}\cdot k^{2}\cdot \log k$. As a main ingredient of the proof of our result, we show that for every graph $H$ on $h$ vertices and pathwidth at most 2, either $G$ contains $k$ disjoint copies of $H$ as a minor or the treewidth of $G$ is upper-bounded by $2^{O(h^2)}\cdot k^{2}\cdot \log k$. We finally prove that the exponential dependence on $h$ in these bounds can be avoided if $H=K_{2,r}$. In particular, we show that $f_{K_{2,r}}=O(r^2\cdot k^2)$