A star-comb lemma for finite digraphs

TL;DR

This work extends the star–comb lemma from undirected graphs to strongly connected digraphs by proving that, for every $n$, a function $g(n)$ exists so that any strongly connected digraph $D$ with a large subset $U$ contains a strongly connected butterfly minor shaped by a star or a comb and having $n$ teeth in $U$. The approach builds a central connected structure (the centre) attached to many $A$--$U$ paths and uses laced paths to simplify two-path witnesses of connectivity. A Ramsey-theoretic, case-based analysis then extracts either a star- or comb-shaped minor with the required teeth, establishing the directed analogue of the finite star–comb phenomenon. The results lay groundwork for understanding unavoidable butterfly minors in directed graphs, with a planned extension to the infinite setting in a subsequent paper.

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.

Figures (5)

• Figure 1: Three directed graphs shaped by a star.
• Figure 2: Two directed graphs shaped by a comb.
• Figure 3: Two laced directed paths.
• Figure 4: Directed graphs of types \ref{['itm:main_structure_1', 'itm:main_structure_2', 'itm:main_structure_3']} in \ref{['propmainstructure']}.
• Figure 5: The setup in the proof of \ref{['propgroundwork']}: A strongly connected subgraph $A$ and two disjoint directed $A$--$U$ paths $P_1$ and $P_2$. The directed paths $Q_1$ and $Q_2$ are laced with the directed paths $P_1$ and $P_2$, respectively, and share a common terminal segment. Furthermore, $Q_1$ avoids $P_2$ and $Q_2$ avoids $P_1$.

Theorems & Definitions (21)

• Theorem 1.1
• Corollary 1.2
• Theorem 2.1
• Proposition 2.3
• proof
• Proposition 3.1
• proof
• Lemma 4.0
• Proposition 4.1
• proof