Erd\H{o}s-P\'osa property of tripods in directed graphs

TL;DR

This paper proves that in directed graphs, the Erdős-Pósa property holds for tripods, meaning either many disjoint tripods exist or a bounded vertex set intersects all tripods, with a specific bounding function.

Contribution

It establishes the Erdős-Pósa property for tripods in directed graphs, a new structural result in directed graph theory.

Findings

Existence of a function f(k) bounding the size of vertex sets intersecting all tripods

Either k disjoint tripods exist or a small vertex set hits all tripods

Extension of Erdős-Pósa property to a new class of subgraphs in directed graphs

Abstract

Let $D$ be a directed graphs with distinguished sets of sources $S\subseteq V(D)$ and sinks $T\subseteq V(D)$. A tripod in $D$ is a subgraph consisting of the union of two $S$-$T$-paths that have distinct start-vertices and the same end-vertex, and are disjoint apart from sharing a suffix. We prove that tripods in directed graphs exhibit the Erd\H{o}s-P\'osa property. More precisely, there is a function $f\colon \mathbb{N}\to \mathbb{N}$ such that for every digraph $D$ with sources $S$ and sinks $T$, if $D$ does not contain $k$ vertex-disjoint tripods, then there is a set of at most $f(k)$ vertices that meets all the tripods in $D$.