Distal combinatorial tools for graphs of bounded twin-width

TL;DR

This paper explores the properties of graphs with bounded twin-width, demonstrating their linear neighborhood complexity and distal nature, which enables the application of advanced combinatorial lemmas for analysis.

Contribution

It establishes that graphs of bounded twin-width have linear neighborhood complexity and are distal, providing new combinatorial tools for their study.

Findings

Graphs of bounded twin-width have linear neighborhood complexity.

Edge relations in such graphs are distal, enabling the use of distal combinatorial lemmas.

Application of distal regularity and cutting lemmas to these graphs.

Abstract

We study set systems formed by neighborhoods in graphs of bounded twin-width. We start by proving that such graphs have linear neighborhood complexity, in analogy to previous results concerning graphs from classes with bounded expansion and of bounded clique-width. Next, we shift our attention to the notions of distality and abstract cell decomposition, which come from model theory. We give a direct combinatorial proof that the edge relation is distal in classes of ordered graphs of bounded twin-width. This allows us to apply Distal cutting lemma and Distal regularity lemma, so we obtain powerful combinatorial tools for graphs of bounded twin-width.