A Star-Comb Lemma for Finite Digraphs

Abstract

It is well-known that for every set $U$ of vertices in a connected graph $G$ there is either a subdivided star in $G$ with a large number of leaves in $U$, or a comb in $G$ with a large number of teeth in $U$.

In this paper we extend this property to directed graphs. More precisely, we prove that for every $n \in \mathbb{N}$ and every sufficiently large set $U$ of vertices in a strongly connected directed graph $D$, there exists a strongly connected butterfly minor of $D$ with $n$ teeth in $U$ that is either 'shaped' by a star or 'shaped' by a comb.