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# Characterising bounded expansion by neighbourhood complexity

Reidl, Felix and Sánchez Villaamil, F. and Stavropoulos, K. (2018) Characterising bounded expansion by neighbourhood complexity. Characterising bounded expansion by neighbourhood complexity 75 , pp. 152-168. ISSN 0195-6698.

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## Abstract

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

Item Type: Article School of Business, Economics & Informatics > Computer Science and Information Systems Felix Reidl 26 Oct 2018 07:04 13 Feb 2021 04:15 https://eprints.bbk.ac.uk/id/eprint/24753