A star-comb lemma for finite digraphs

TL;DR

This paper extends a classical undirected graph property to directed graphs, showing that large vertex sets in strongly connected digraphs contain specific star- or comb-shaped butterfly minors.

Contribution

It introduces a directed analogue of the star-comb lemma, demonstrating the existence of particular butterfly minors in strongly connected digraphs with large vertex sets.

Findings

Existence of star-shaped butterfly minors with many leaves in large vertex sets.

Existence of comb-shaped butterfly minors with many teeth in large vertex sets.

Extension of the undirected star-comb lemma to directed graphs.

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.