{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:FL53I4ITONLDFVZRFVY3KVQ4C3","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"c906e1cb85a2f677032e0faf0e57c2301244f5e5e79e47e3edc7ce05b81c0428","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.DM","submitted_at":"2026-07-18T17:11:16Z","title_canon_sha256":"0477da420d4a68435383e175ea462cc082336c0e69c9de2785b206bd12ad4e4a"},"schema_version":"1.0","source":{"id":"2607.16889","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.16889","created_at":"2026-07-21T01:21:04Z"},{"alias_kind":"arxiv_version","alias_value":"2607.16889v1","created_at":"2026-07-21T01:21:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.16889","created_at":"2026-07-21T01:21:04Z"},{"alias_kind":"pith_short_12","alias_value":"FL53I4ITONLD","created_at":"2026-07-21T01:21:04Z"},{"alias_kind":"pith_short_16","alias_value":"FL53I4ITONLDFVZR","created_at":"2026-07-21T01:21:04Z"},{"alias_kind":"pith_short_8","alias_value":"FL53I4IT","created_at":"2026-07-21T01:21:04Z"}],"graph_snapshots":[{"event_id":"sha256:5bbc91384130393e9937a8b8bb88a327de2ea06000f081a1bda2e1ab37901c3f","target":"graph","created_at":"2026-07-21T01:21:04Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2607.16889/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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, $nc(G,k)\\in O(k)$ ","authors_text":"Aline Parreau, Florent Foucaud, Ga\\'etan Berthe, Tuomo Lehtil\\\"a","cross_cats":["math.CO"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.DM","submitted_at":"2026-07-18T17:11:16Z","title":"Neighbourhood complexity and identification problems for graphs of bounded treewidth and pathwidth"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.16889","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:c245803f05bb014d121dec5d4718b8b05f18a699ed5fbc8832696457ab20a067","target":"record","created_at":"2026-07-21T01:21:04Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"c906e1cb85a2f677032e0faf0e57c2301244f5e5e79e47e3edc7ce05b81c0428","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.DM","submitted_at":"2026-07-18T17:11:16Z","title_canon_sha256":"0477da420d4a68435383e175ea462cc082336c0e69c9de2785b206bd12ad4e4a"},"schema_version":"1.0","source":{"id":"2607.16889","kind":"arxiv","version":1}},"canonical_sha256":"2afbb47113735632d7312d71b5561c16ec0d94a8b6caca993a96c96f85123520","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2afbb47113735632d7312d71b5561c16ec0d94a8b6caca993a96c96f85123520","first_computed_at":"2026-07-21T01:21:04.060566Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-21T01:21:04.060566Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bBekwQqraVdYSzLlYRxLIF9ociXZZ4yOARym3sRxTcUJZyrZTYhP8OZgqRZRgEs00jl+7NLDa1n6ki7n14XGAQ==","signature_status":"signed_v1","signed_at":"2026-07-21T01:21:04.061429Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.16889","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c245803f05bb014d121dec5d4718b8b05f18a699ed5fbc8832696457ab20a067","sha256:5bbc91384130393e9937a8b8bb88a327de2ea06000f081a1bda2e1ab37901c3f"],"state_sha256":"514afdeda50c0dc195780b12aea1259248cdedc3a084093427a7334297188e6a"}