A star-comb lemma for infinite digraphs

TL;DR

This paper extends the star-comb lemma from undirected to directed graphs, showing that infinite strongly connected digraphs contain specific infinite structures with vertices in a given set.

Contribution

It introduces a directed analogue of the star-comb lemma, identifying new infinite structures in strongly connected digraphs.

Findings

Existence of a strongly connected butterfly minor with infinitely many vertices in U

Presence of structures shaped like a star, a comb, or a chain of triangles

Applicable to infinite strongly connected directed graphs

Abstract

The star-comb lemma is a standard tool in infinite graph theory, which states that for every infinite set $U$ of vertices in a connected graph $G$ there exists either a subdivided infinite star in $G$ with all leaves in $U$, or an infinite comb in $G$ with all teeth in $U$. In this paper, we elaborate a counterpart of the star-comb lemma for directed graphs. More precisely, we prove that for every infinite set $U$ of vertices in a strongly connected directed graph $D$, there exists a strongly connected butterfly minor of $D$ with infinitely many teeth in $U$ that is either shaped by a star or shaped by a comb, or is a chain of triangles.