Characterising bounded expansion by neighbourhood complexity

We show that a graph class 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 any G∈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 r-centred colouring number and its weakr-colouring number, which provide known characterisations to the property of bounded expansion.