Hitting minors, subdivisions, and immersions in tournaments

TL;DR

This paper extends the Erdős-Pósa property to directed graphs within tournaments, showing that certain minors and immersions have the property, which relates to covering and packing parameters in graph theory.

Contribution

It demonstrates that classes of directed graphs containing a fixed strongly-connected graph as a minor or immersion have the Erdős-Pósa property in tournaments, using recent structural results.

Findings

Directed minors and immersions have the Erdős-Pósa property in tournaments.

Vertex-Erdős-Pósa property holds for strong minors, butterfly minors, and topological minors.

Edge-Erdős-Pósa property holds for immersions of strongly-connected graphs.

Abstract

The Erd\H{o}s-P\'osa property relates parameters of covering and packing of combinatorial structures and has been mostly studied in the setting of undirected graphs. In this note, we use results of Chudnovsky, Fradkin, Kim, and Seymour to show that, for every directed graph $H$ (resp. strongly-connected directed graph $H$), the class of directed graphs that contain $H$ as a strong minor (resp. butterfly minor, topological minor) has the vertex-Erd\H{o}s-P\'osa property in the class of tournaments. We also prove that if $H$ is a strongly-connected directed graph, the class of directed graphs containing $H$ as an immersion has the edge-Erd\H{o}s-P\'osa property in the class of tournaments.