{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:JB6774LVW3VH3LLYC46V252OP2","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":"d6b6467b16dc008eeb294285cd458ec7014237184cf61b72f5326721dd8e3fcc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-05-18T07:20:19Z","title_canon_sha256":"68dc6dc6cc6e9a471aa6c352940175f5ae0e3367930be4347c26d061bddb73c6"},"schema_version":"1.0","source":{"id":"1105.3549","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1105.3549","created_at":"2026-05-18T02:21:59Z"},{"alias_kind":"arxiv_version","alias_value":"1105.3549v2","created_at":"2026-05-18T02:21:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.3549","created_at":"2026-05-18T02:21:59Z"},{"alias_kind":"pith_short_12","alias_value":"JB6774LVW3VH","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_16","alias_value":"JB6774LVW3VH3LLY","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_8","alias_value":"JB6774LV","created_at":"2026-05-18T12:26:32Z"}],"graph_snapshots":[{"event_id":"sha256:cb41e8af4431d6b3d3a305428bc735bdb5dea88f4a62dc091e4c312ab12aa7ce","target":"graph","created_at":"2026-05-18T02:21:59Z","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":"The graph minor structure theorem by Robertson and Seymour shows that every graph that excludes a fixed minor can be constructed by a combination of four ingredients: graphs embedded in a surface of bounded genus, a bounded number of vortices of bounded width, a bounded number of apex vertices, and the clique-sum operation. This paper studies the converse question: What is the maximum order of a complete graph minor in a graph constructed using these four ingredients? Our main result answers this question up to a constant factor.","authors_text":"David R. Wood, Gwena\\\"el Joret","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-05-18T07:20:19Z","title":"Complete Graph Minors and the Graph Minor Structure Theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.3549","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":"sha256:4a352a36c30374585e1b2187ddff56f41bbbb205000e43ae47c0f7809a413382","target":"record","created_at":"2026-05-18T02:21:59Z","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":"d6b6467b16dc008eeb294285cd458ec7014237184cf61b72f5326721dd8e3fcc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-05-18T07:20:19Z","title_canon_sha256":"68dc6dc6cc6e9a471aa6c352940175f5ae0e3367930be4347c26d061bddb73c6"},"schema_version":"1.0","source":{"id":"1105.3549","kind":"arxiv","version":2}},"canonical_sha256":"487dfff175b6ea7dad78173d5d774e7ebf9c33beed48eb75c57f32bf4875f985","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"487dfff175b6ea7dad78173d5d774e7ebf9c33beed48eb75c57f32bf4875f985","first_computed_at":"2026-05-18T02:21:59.478812Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:21:59.478812Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YHQiQsESMI2rGRPsEQtDsWDPd5teYlVyV3TsxxrcuMKHpEQ58SL3elEaQzeYrUweP/pm/pwaMT0/n34WpbG5DA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:21:59.479512Z","signed_message":"canonical_sha256_bytes"},"source_id":"1105.3549","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4a352a36c30374585e1b2187ddff56f41bbbb205000e43ae47c0f7809a413382","sha256:cb41e8af4431d6b3d3a305428bc735bdb5dea88f4a62dc091e4c312ab12aa7ce"],"state_sha256":"81bd90beb885638e45b98d04c9e26ea0a2ddd9fd33b3351b8154f7886b32f03b"}