{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:JAW3NNVAF7PG4IFLJQY7EE24UL","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":"db2c5b2520e743a954855d80412acc6d5a37e31a3afb01633aa11c0811b1b484","cross_cats_sorted":["cs.DM"],"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.CO","submitted_at":"2011-02-28T20:13:21Z","title_canon_sha256":"ba3ab674c4ef90a4c30250f85ee2832044b0cb903a39c496d68a38ada02674dd"},"schema_version":"1.0","source":{"id":"1102.5762","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1102.5762","created_at":"2026-05-18T04:27:46Z"},{"alias_kind":"arxiv_version","alias_value":"1102.5762v1","created_at":"2026-05-18T04:27:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.5762","created_at":"2026-05-18T04:27:46Z"},{"alias_kind":"pith_short_12","alias_value":"JAW3NNVAF7PG","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_16","alias_value":"JAW3NNVAF7PG4IFL","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_8","alias_value":"JAW3NNVA","created_at":"2026-05-18T12:26:32Z"}],"graph_snapshots":[{"event_id":"sha256:9ffd7b414abec8317d60327e9b4eb261d48188c62aee762815a5c770f43c9632","target":"graph","created_at":"2026-05-18T04:27:46Z","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":"One of the major results of [N. Robertson and P. D. Seymour. Graph minors. XIII. The disjoint paths problem. J. Combin. Theory Ser. B, 63(1):65--110, 1995], also known as the weak structure theorem, revealed the local structure of graphs excluding some graph as a minor: each such graph $G$ either has small treewidth or contains the subdivision of a wall that can be arranged \"bidimensionally\" inside $G$, given that some small set of vertices are removed. We prove an optimized version of that theorem where (i) the relation between the treewidth of the graph and the height of the wall is linear (","authors_text":"Archontia C. Giannopoulou, Dimitrios M. Thilikos","cross_cats":["cs.DM"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.CO","submitted_at":"2011-02-28T20:13:21Z","title":"Optimizing the Graph Minors Weak Structure Theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.5762","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":"sha256:436b8d3f6d711e8518d25c687bc40aff42b5a955bedd84a6e10a7c1dae2df9f6","target":"record","created_at":"2026-05-18T04:27:46Z","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":"db2c5b2520e743a954855d80412acc6d5a37e31a3afb01633aa11c0811b1b484","cross_cats_sorted":["cs.DM"],"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.CO","submitted_at":"2011-02-28T20:13:21Z","title_canon_sha256":"ba3ab674c4ef90a4c30250f85ee2832044b0cb903a39c496d68a38ada02674dd"},"schema_version":"1.0","source":{"id":"1102.5762","kind":"arxiv","version":1}},"canonical_sha256":"482db6b6a02fde6e20ab4c31f2135ca2f1f18adc739c9d2d3de3d60fc4093118","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"482db6b6a02fde6e20ab4c31f2135ca2f1f18adc739c9d2d3de3d60fc4093118","first_computed_at":"2026-05-18T04:27:46.032750Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:27:46.032750Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Hg1uqMKGVA6OJAQpprTcAF92FaqKwlnNQlEO0YUjNV6LOOh6EQ14p897qi9NNrL3CGQtm9YBXXD11tDfl+sjBw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:27:46.033298Z","signed_message":"canonical_sha256_bytes"},"source_id":"1102.5762","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:436b8d3f6d711e8518d25c687bc40aff42b5a955bedd84a6e10a7c1dae2df9f6","sha256:9ffd7b414abec8317d60327e9b4eb261d48188c62aee762815a5c770f43c9632"],"state_sha256":"09e9857cfe18f3d8e62cf6a9f1024c467e99b586ba4837550f4d8f583937b1bb"}