{"paper":{"title":"Neighbourhood complexity and identification problems for graphs of bounded treewidth and pathwidth","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Aline Parreau, Florent Foucaud, Ga\\'etan Berthe, Tuomo Lehtil\\\"a","submitted_at":"2026-07-18T17:11:16Z","abstract_excerpt":"The neighbourhood complexity $nc(G,k)$ of a graph $G$ is a quantity measuring, for a graph $G$ and an integer $k$, the maximum possible number (over all vertex subsets $S$ of size $k$) $|\\{N[v]\\cap S, v\\in V(G)\\}|$ of $S$-neighbourhoods in $G$. This notion is important in structural graph theory and algorithm design (especially in parameterized complexity, in particular model checking and kernelization).\n  While generally $nc(G,k)\\leq 2^k$ and this bound can be achieved, it is known that sparse graphs and structured dense graphs have linear neighbourhood complexity, that is, $nc(G,k)\\in O(k)$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.16889","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.16889/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}