Characterising Bounded Expansion by Neighbourhood Complexity

TL;DR

This paper characterizes bounded expansion in graph classes through the concept of neighbourhood complexity, linking it to known parameters like colouring numbers, and providing a new perspective on graph sparsity.

Contribution

It establishes an equivalence between bounded expansion and bounded r-neighbourhood complexity, connecting it to existing graph parameters.

Findings

Bounded expansion is characterized by linear r-neighbourhood complexity.

The paper bounds r-neighbourhood complexity using r-centred colouring and weak r-colouring numbers.

Provides a new characterization of sparse graph classes based on neighbourhood complexity.

Abstract

We show that a graph class $\cal G$ has bounded expansion if and only if it has bounded $r$-neighbourhood complexity, i.e. for any vertex set $X$ of any subgraph $H$ of $G\in\cal G$, the number of subsets of $X$ which are exact $r$-neighbourhoods of vertices of $H$ on $X$ is linear to the size of $X$. This is established by bounding the $r$-neighbourhood complexity of a graph in terms of both its $r$-centred colouring number and its weak $r$-colouring number, which provide known characterisations to the property of bounded expansion.