{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2004:FEVTYVIX72YHID5SC7LU4SRDH6","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":"46b4be0e861b569a90971e2d8a2e596b60c523efda6d3494b203f4b8640165b2","cross_cats_sorted":["cs.DM"],"license":"","primary_cat":"math.CO","submitted_at":"2004-06-08T04:22:43Z","title_canon_sha256":"992e946ab8c92e389d246270fd32f5caf74f8367d0633eb120fecc9db711aa9d"},"schema_version":"1.0","source":{"id":"math/0406140","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0406140","created_at":"2026-05-18T04:32:43Z"},{"alias_kind":"arxiv_version","alias_value":"math/0406140v5","created_at":"2026-05-18T04:32:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0406140","created_at":"2026-05-18T04:32:43Z"},{"alias_kind":"pith_short_12","alias_value":"FEVTYVIX72YH","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_16","alias_value":"FEVTYVIX72YHID5S","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_8","alias_value":"FEVTYVIX","created_at":"2026-05-18T12:25:52Z"}],"graph_snapshots":[{"event_id":"sha256:0767e099330dc0055675e8d30267eddd97f7988a74093f5e55a40a83b5582500","target":"graph","created_at":"2026-05-18T04:32:43Z","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 the class F of 2-connected non-planar K_{3,3}-subdivision-free graphs that are embeddable in the projective plane. We show that these graphs admit a unique decomposition as a graph K_5 (the core) where the edges are replaced by two-pole networks constructed from 2-connected planar graphs. A method to enumerate these graphs in the labelled case is described. Moreover, we enumerate the homeomorphically irreducible graphs in F and homeomorphically irreducible 2-connected planar graphs. Particular use is made of two-pole directed series-parallel networks. We also show that the number m","authors_text":"Andrei Gagarin, Gilbert Labelle, Pierre Leroux (LACIM, Universite du Quebec a Montreal)","cross_cats":["cs.DM"],"headline":"","license":"","primary_cat":"math.CO","submitted_at":"2004-06-08T04:22:43Z","title":"The structure and labelled enumeration of K_{3,3}-subdivision-free projective-planar graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0406140","kind":"arxiv","version":5},"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":"sha256:9145e0e544fde87e17af9aa15c18fa9a24f0e4627bf768e664bd4ba1b974ab29","target":"record","created_at":"2026-05-18T04:32:43Z","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":"46b4be0e861b569a90971e2d8a2e596b60c523efda6d3494b203f4b8640165b2","cross_cats_sorted":["cs.DM"],"license":"","primary_cat":"math.CO","submitted_at":"2004-06-08T04:22:43Z","title_canon_sha256":"992e946ab8c92e389d246270fd32f5caf74f8367d0633eb120fecc9db711aa9d"},"schema_version":"1.0","source":{"id":"math/0406140","kind":"arxiv","version":5}},"canonical_sha256":"292b3c5517feb0740fb217d74e4a233f894799860db1e21be49dfd0c11181fd2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"292b3c5517feb0740fb217d74e4a233f894799860db1e21be49dfd0c11181fd2","first_computed_at":"2026-05-18T04:32:43.998392Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:32:43.998392Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LdcjEn7cQ7/0W40u06Yi3ebO3A9loYBRxNe603Tu0GEy1yA6RlTZV2ih7V2Krt9bNXY5T5zVDHsETZubA/EhDA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:32:43.998862Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0406140","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9145e0e544fde87e17af9aa15c18fa9a24f0e4627bf768e664bd4ba1b974ab29","sha256:0767e099330dc0055675e8d30267eddd97f7988a74093f5e55a40a83b5582500"],"state_sha256":"725116a847d34cc7d291205d194320eda7c0cba6d0ee050d3d0a6f7e62d093c2"}