{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:GR5DT4NJUZR2BTSLLW7H3ACSDM","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":"b8ecf1817aebf0f12635947efe579e78b20a72de7b4b1f6858afa09270563251","cross_cats_sorted":["cs.CG","cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-06-14T11:17:47Z","title_canon_sha256":"b655d998b0ad8da65f5f4c1b63575e8e157a6ac03e2932bd3740c3f64ca7262a"},"schema_version":"1.0","source":{"id":"1506.04380","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.04380","created_at":"2026-05-18T00:40:14Z"},{"alias_kind":"arxiv_version","alias_value":"1506.04380v2","created_at":"2026-05-18T00:40:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.04380","created_at":"2026-05-18T00:40:14Z"},{"alias_kind":"pith_short_12","alias_value":"GR5DT4NJUZR2","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_16","alias_value":"GR5DT4NJUZR2BTSL","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_8","alias_value":"GR5DT4NJ","created_at":"2026-05-18T12:29:22Z"}],"graph_snapshots":[{"event_id":"sha256:7d427985c886baff2e00960efa858305c7887d48d0fe7aa80b60abf7a45859df","target":"graph","created_at":"2026-05-18T00:40:14Z","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"},"paper":{"abstract_excerpt":"We consider relations between the size, treewidth, and local crossing number (maximum number of crossings per edge) of graphs embedded on topological surfaces. We show that an $n$-vertex graph embedded on a surface of genus $g$ with at most $k$ crossings per edge has treewidth $O(\\sqrt{(g+1)(k+1)n})$ and layered treewidth $O((g+1)k)$, and that these bounds are tight up to a constant factor. As a special case, the $k$-planar graphs with $n$ vertices have treewidth $O(\\sqrt{(k+1)n})$ and layered treewidth $O(k+1)$, which are tight bounds that improve a previously known $O((k+1)^{3/4}n^{1/2})$ tr","authors_text":"David Eppstein, David R. Wood, Vida Dujmovi\\'c","cross_cats":["cs.CG","cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-06-14T11:17:47Z","title":"Structure of Graphs with Locally Restricted Crossings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.04380","kind":"arxiv","version":2},"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":"sha256:46177a27bd3334d36bc933ccd18d9f18e43628c7d064bdd66f6f3a9d93e504a9","target":"record","created_at":"2026-05-18T00:40:14Z","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":"b8ecf1817aebf0f12635947efe579e78b20a72de7b4b1f6858afa09270563251","cross_cats_sorted":["cs.CG","cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-06-14T11:17:47Z","title_canon_sha256":"b655d998b0ad8da65f5f4c1b63575e8e157a6ac03e2932bd3740c3f64ca7262a"},"schema_version":"1.0","source":{"id":"1506.04380","kind":"arxiv","version":2}},"canonical_sha256":"347a39f1a9a663a0ce4b5dbe7d80521b1283d87158b8a2138b3a7d2ae754db12","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"347a39f1a9a663a0ce4b5dbe7d80521b1283d87158b8a2138b3a7d2ae754db12","first_computed_at":"2026-05-18T00:40:14.239725Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:40:14.239725Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UorGEVYipf/Pecj3B9oM9v43gbJDZQ6Bl9sMJURA5utVQiGjghHmLAExBCu83usnm5Ma8vsRpLQAJnUic1tDDw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:40:14.240429Z","signed_message":"canonical_sha256_bytes"},"source_id":"1506.04380","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:46177a27bd3334d36bc933ccd18d9f18e43628c7d064bdd66f6f3a9d93e504a9","sha256:7d427985c886baff2e00960efa858305c7887d48d0fe7aa80b60abf7a45859df"],"state_sha256":"7bd555df5453119a2471eba8d8833133927a7a3f95e83e162090871f4720715b"}