{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:G473JKC6KEFX7NCZLW4SLYCGL7","short_pith_number":"pith:G473JKC6","canonical_record":{"source":{"id":"0912.4778","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-12-24T04:20:08Z","cross_cats_sorted":[],"title_canon_sha256":"c7a7b1766aab7bb9f3b50bd435189988c9518df877224e2bce05453dad11b234","abstract_canon_sha256":"dad85608102b1998eda06ed0c61f370ebf6a45ade757bceef3cebebff3890075"},"schema_version":"1.0"},"canonical_sha256":"373fb4a85e510b7fb4595db925e0465fe53154e4ed793d4131549d43d8b67d03","source":{"kind":"arxiv","id":"0912.4778","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0912.4778","created_at":"2026-05-18T04:36:33Z"},{"alias_kind":"arxiv_version","alias_value":"0912.4778v2","created_at":"2026-05-18T04:36:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0912.4778","created_at":"2026-05-18T04:36:33Z"},{"alias_kind":"pith_short_12","alias_value":"G473JKC6KEFX","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_16","alias_value":"G473JKC6KEFX7NCZ","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_8","alias_value":"G473JKC6","created_at":"2026-05-18T12:25:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:G473JKC6KEFX7NCZLW4SLYCGL7","target":"record","payload":{"canonical_record":{"source":{"id":"0912.4778","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-12-24T04:20:08Z","cross_cats_sorted":[],"title_canon_sha256":"c7a7b1766aab7bb9f3b50bd435189988c9518df877224e2bce05453dad11b234","abstract_canon_sha256":"dad85608102b1998eda06ed0c61f370ebf6a45ade757bceef3cebebff3890075"},"schema_version":"1.0"},"canonical_sha256":"373fb4a85e510b7fb4595db925e0465fe53154e4ed793d4131549d43d8b67d03","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:36:33.953867Z","signature_b64":"ki6bV6rDs1hu+5aJa70ickTsUUksgFHrf1wat3PT3tYsy79XTaDCDfB5RUsJQ1LCFlYufavPYuz7njjXQIlCBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"373fb4a85e510b7fb4595db925e0465fe53154e4ed793d4131549d43d8b67d03","last_reissued_at":"2026-05-18T04:36:33.953464Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:36:33.953464Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0912.4778","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:36:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2IOmv4nXrKKPfzUP+o3CMXHTF44j+2fGqxrKfYk6skdFqDIK7W0914g8cWjIqchuw4I4Ja+H4alDfpNNNTO5Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T19:15:57.157068Z"},"content_sha256":"ff7b16e3161822997b54a11f8b9c7834b22233992a3bc5eddb143ffd699a2d86","schema_version":"1.0","event_id":"sha256:ff7b16e3161822997b54a11f8b9c7834b22233992a3bc5eddb143ffd699a2d86"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:G473JKC6KEFX7NCZLW4SLYCGL7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Large Non-Planar Graphs and an Application to Crossing-Critical Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bogdan Oporowski, Dirk Vertigan, Guoli Ding, Robin Thomas","submitted_at":"2009-12-24T04:20:08Z","abstract_excerpt":"We prove that, for every positive integer k, there is an integer N such that every 4-connected non-planar graph with at least N vertices has a minor isomorphic to K_{4,k}, the graph obtained from a cycle of length 2k+1 by adding an edge joining every pair of vertices at distance exactly k, or the graph obtained from a cycle of length k by adding two vertices adjacent to each other and to every vertex on the cycle. We also prove a version of this for subdivisions rather than minors, and relax the connectivity to allow 3-cuts with one side planar and of bounded size. We deduce that for every int"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0912.4778","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:36:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lHdnfd4HsIz1XJsGkOxAw1PWq0PHFEwAA9pTTRg07Jh/W9fa6TNGKPMjqDH1F2pOIN89NhxjmtBmhE5K0Ye1Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T19:15:57.157452Z"},"content_sha256":"d7b7fc0fcbbe5a5f856c2c275149b8f201718fdb09a81120c210c3b1e35bfc20","schema_version":"1.0","event_id":"sha256:d7b7fc0fcbbe5a5f856c2c275149b8f201718fdb09a81120c210c3b1e35bfc20"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/G473JKC6KEFX7NCZLW4SLYCGL7/bundle.json","state_url":"https://pith.science/pith/G473JKC6KEFX7NCZLW4SLYCGL7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/G473JKC6KEFX7NCZLW4SLYCGL7/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-21T19:15:57Z","links":{"resolver":"https://pith.science/pith/G473JKC6KEFX7NCZLW4SLYCGL7","bundle":"https://pith.science/pith/G473JKC6KEFX7NCZLW4SLYCGL7/bundle.json","state":"https://pith.science/pith/G473JKC6KEFX7NCZLW4SLYCGL7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/G473JKC6KEFX7NCZLW4SLYCGL7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:G473JKC6KEFX7NCZLW4SLYCGL7","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":"dad85608102b1998eda06ed0c61f370ebf6a45ade757bceef3cebebff3890075","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-12-24T04:20:08Z","title_canon_sha256":"c7a7b1766aab7bb9f3b50bd435189988c9518df877224e2bce05453dad11b234"},"schema_version":"1.0","source":{"id":"0912.4778","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0912.4778","created_at":"2026-05-18T04:36:33Z"},{"alias_kind":"arxiv_version","alias_value":"0912.4778v2","created_at":"2026-05-18T04:36:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0912.4778","created_at":"2026-05-18T04:36:33Z"},{"alias_kind":"pith_short_12","alias_value":"G473JKC6KEFX","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_16","alias_value":"G473JKC6KEFX7NCZ","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_8","alias_value":"G473JKC6","created_at":"2026-05-18T12:25:59Z"}],"graph_snapshots":[{"event_id":"sha256:d7b7fc0fcbbe5a5f856c2c275149b8f201718fdb09a81120c210c3b1e35bfc20","target":"graph","created_at":"2026-05-18T04:36:33Z","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 prove that, for every positive integer k, there is an integer N such that every 4-connected non-planar graph with at least N vertices has a minor isomorphic to K_{4,k}, the graph obtained from a cycle of length 2k+1 by adding an edge joining every pair of vertices at distance exactly k, or the graph obtained from a cycle of length k by adding two vertices adjacent to each other and to every vertex on the cycle. We also prove a version of this for subdivisions rather than minors, and relax the connectivity to allow 3-cuts with one side planar and of bounded size. We deduce that for every int","authors_text":"Bogdan Oporowski, Dirk Vertigan, Guoli Ding, Robin Thomas","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-12-24T04:20:08Z","title":"Large Non-Planar Graphs and an Application to Crossing-Critical Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0912.4778","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:ff7b16e3161822997b54a11f8b9c7834b22233992a3bc5eddb143ffd699a2d86","target":"record","created_at":"2026-05-18T04:36:33Z","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":"dad85608102b1998eda06ed0c61f370ebf6a45ade757bceef3cebebff3890075","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-12-24T04:20:08Z","title_canon_sha256":"c7a7b1766aab7bb9f3b50bd435189988c9518df877224e2bce05453dad11b234"},"schema_version":"1.0","source":{"id":"0912.4778","kind":"arxiv","version":2}},"canonical_sha256":"373fb4a85e510b7fb4595db925e0465fe53154e4ed793d4131549d43d8b67d03","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"373fb4a85e510b7fb4595db925e0465fe53154e4ed793d4131549d43d8b67d03","first_computed_at":"2026-05-18T04:36:33.953464Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:36:33.953464Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ki6bV6rDs1hu+5aJa70ickTsUUksgFHrf1wat3PT3tYsy79XTaDCDfB5RUsJQ1LCFlYufavPYuz7njjXQIlCBA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:36:33.953867Z","signed_message":"canonical_sha256_bytes"},"source_id":"0912.4778","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ff7b16e3161822997b54a11f8b9c7834b22233992a3bc5eddb143ffd699a2d86","sha256:d7b7fc0fcbbe5a5f856c2c275149b8f201718fdb09a81120c210c3b1e35bfc20"],"state_sha256":"639875702167a813f7ce3a3287d8e2b4e00498072501eb3b5e6caa5439a20dda"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lKs96brdW/B/mxGiKKzczj9itWlB/bLzz2nEa6JSYFWOimNFH8mRKOOPKRFOjlldNroeYrsCQuSRTCqYoboHBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-21T19:15:57.159407Z","bundle_sha256":"1453123b8585b32ddb6276221d0f4d87be0210bfcc30389d3bdd784a842b95b5"}}