On the Erd\H{o}s-P\'osa property for immersions and topological minors in tournaments

TL;DR

This paper establishes bounds for the Erdős-Pósa property concerning immersions and topological minors in tournaments, extending previous results to a broader class of digraphs.

Contribution

It generalizes the Erdős-Pósa property to all simple digraphs in tournaments, providing explicit bounds for arc and vertex sets.

Findings

Existence of O_H(k^3) arc sets intersecting all immersions of H

Existence of O_H(k log k) vertex sets intersecting all topological minors of H

Improves previous bounds by Raymond for strongly connected H

Abstract

We consider the Erd\H{o}s-P\'osa property for immersions and topological minors in tournaments. We prove that for every simple digraph $H$, $k\in \mathbb{N}$, and tournament $T$, the following statements hold: (i) If in $T$ one cannot find $k$ arc-disjoint immersion copies of $H$, then there exists a set of $\mathcal{O}_H(k^3)$ arcs that intersects all immersion copies of $H$ in $T$. (ii) If in $T$ one cannot find $k$ vertex-disjoint topological minor copies of $H$, then there exists a set of $\mathcal{O}_H(k\log k)$ vertices that intersects all topological minor copies of $H$ in $T$. This improves the results of Raymond [DMTCS '18], who proved similar statements under the assumption that $H$ is strongly connected.