{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:ZWUPPOJQGCNV2TUTGS5AILA7M6","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":"5331d123731668a3d985c0f9d18a763f03075aefee305c585493a7124e517676","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-10-30T10:52:29Z","title_canon_sha256":"2a597ca985cd7136d6c8886d98bb14f7a767b6805d60867fd931bf930ea8c5c2"},"schema_version":"1.0","source":{"id":"1210.7960","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.7960","created_at":"2026-05-18T02:49:39Z"},{"alias_kind":"arxiv_version","alias_value":"1210.7960v1","created_at":"2026-05-18T02:49:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.7960","created_at":"2026-05-18T02:49:39Z"},{"alias_kind":"pith_short_12","alias_value":"ZWUPPOJQGCNV","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_16","alias_value":"ZWUPPOJQGCNV2TUT","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_8","alias_value":"ZWUPPOJQ","created_at":"2026-05-18T12:27:30Z"}],"graph_snapshots":[{"event_id":"sha256:d7aaa169b8256b4048bb91c7b677e0cf1f84ec003c5aecd15e9c7f25423b8120","target":"graph","created_at":"2026-05-18T02:49:39Z","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":"This paper studies a class of binomial ideals associated to graphs with finite vertex sets. They generalize the binomial edge ideals, and they arise in the study of conditional independence ideals. A Gr\\\"obner basis can be computed by studying paths in the graph. Since these Gr\\\"obner bases are square-free, generalized binomial edge ideals are radical. To find the primary decomposition a combinatorial problem involving the connected components of subgraphs has to be solved. The irreducible components of the solution variety are all rational.","authors_text":"Johannes Rauh","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-10-30T10:52:29Z","title":"Generalized Binomial Edge Ideals"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.7960","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":"sha256:25bc1a924701f87cb0a2cb4ed4eb9231042806fb3f4c1c193a6f918d589969e3","target":"record","created_at":"2026-05-18T02:49:39Z","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":"5331d123731668a3d985c0f9d18a763f03075aefee305c585493a7124e517676","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-10-30T10:52:29Z","title_canon_sha256":"2a597ca985cd7136d6c8886d98bb14f7a767b6805d60867fd931bf930ea8c5c2"},"schema_version":"1.0","source":{"id":"1210.7960","kind":"arxiv","version":1}},"canonical_sha256":"cda8f7b930309b5d4e9334ba042c1f67a077d2a9f3ed57cbeb4a6a16f0a99bc9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cda8f7b930309b5d4e9334ba042c1f67a077d2a9f3ed57cbeb4a6a16f0a99bc9","first_computed_at":"2026-05-18T02:49:39.100738Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:49:39.100738Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XpzcUh0douaCl8y+pkDgRshOHUbh+3ozdvC2pgsbkADfFY0dU4p7wZj4f7EtD0wcjfL9sNtXD1eH8NyCJosZDg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:49:39.101079Z","signed_message":"canonical_sha256_bytes"},"source_id":"1210.7960","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:25bc1a924701f87cb0a2cb4ed4eb9231042806fb3f4c1c193a6f918d589969e3","sha256:d7aaa169b8256b4048bb91c7b677e0cf1f84ec003c5aecd15e9c7f25423b8120"],"state_sha256":"de9a3afc91a2b5a82c178fff917501696f6fcab2daf5748eb6770e776411d0e8"}