{"paper":{"title":"Characterising Bounded Expansion by Neighbourhood Complexity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Felix Reidl, Fernando S\\'anchez Villaamil, Konstantinos Stavropoulos","submitted_at":"2016-03-31T11:34:48Z","abstract_excerpt":"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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.09532","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}