{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:KAQBMFMZDM4AMQRYPZOAXY62QB","short_pith_number":"pith:KAQBMFMZ","canonical_record":{"source":{"id":"1304.4211","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-04-15T19:26:45Z","cross_cats_sorted":[],"title_canon_sha256":"488701670107c2d48cfc0a4fa8e9b15abe2abd7e52bc40c445f8d7dc301acb92","abstract_canon_sha256":"18353431fc6536af6f4b96fb2862f54b7f626f9477b87a5fe379767c8dd6ace1"},"schema_version":"1.0"},"canonical_sha256":"50201615991b380642387e5c0be3da8056b1073738afdb230542141b3f551b20","source":{"kind":"arxiv","id":"1304.4211","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.4211","created_at":"2026-05-18T02:58:32Z"},{"alias_kind":"arxiv_version","alias_value":"1304.4211v1","created_at":"2026-05-18T02:58:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.4211","created_at":"2026-05-18T02:58:32Z"},{"alias_kind":"pith_short_12","alias_value":"KAQBMFMZDM4A","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_16","alias_value":"KAQBMFMZDM4AMQRY","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_8","alias_value":"KAQBMFMZ","created_at":"2026-05-18T12:27:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:KAQBMFMZDM4AMQRYPZOAXY62QB","target":"record","payload":{"canonical_record":{"source":{"id":"1304.4211","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-04-15T19:26:45Z","cross_cats_sorted":[],"title_canon_sha256":"488701670107c2d48cfc0a4fa8e9b15abe2abd7e52bc40c445f8d7dc301acb92","abstract_canon_sha256":"18353431fc6536af6f4b96fb2862f54b7f626f9477b87a5fe379767c8dd6ace1"},"schema_version":"1.0"},"canonical_sha256":"50201615991b380642387e5c0be3da8056b1073738afdb230542141b3f551b20","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:58:32.622399Z","signature_b64":"tdLQrvSZpedN/hln2N38KyAuRvxYzxp0DbZIHxfI/rDSZN3gt8CBISA1XJuN14uVHPwWTuCSeGAWhRxcuoFqDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"50201615991b380642387e5c0be3da8056b1073738afdb230542141b3f551b20","last_reissued_at":"2026-05-18T02:58:32.621767Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:58:32.621767Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1304.4211","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:58:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WBj5UycbE/eUiYV2b7e4OFZBQqZTD7XtzCXCdXT9471e2tcuL62rX18+5unGWcHKJIsAESbZ3v0T2wuXJYsSBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T12:01:24.409869Z"},"content_sha256":"30fb165fa1ec26407475d169161242bd80f1306f5a3a3ab5baab54743e3ee15a","schema_version":"1.0","event_id":"sha256:30fb165fa1ec26407475d169161242bd80f1306f5a3a3ab5baab54743e3ee15a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:KAQBMFMZDM4AMQRYPZOAXY62QB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Graphs with two trivial critical ideals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Carlos A. Alfaro, Carlos E. Valencia","submitted_at":"2013-04-15T19:26:45Z","abstract_excerpt":"The critical ideals of a graph are the determinantal ideals of the generalized Laplacian matrix associated to a graph. A basic property of the critical ideals of graphs asserts that the graphs with at most k trivial critical ideals, $\\Gamma_{\\leq k}$, are closed under induced subgraphs. In this article we find the set of minimal forbidden subgraphs for $\\Gamma_{\\leq 2}$, and we use this forbidden subgraphs to get a classification of the graphs in $\\Gamma_{\\leq 2}$. As a consequence we give a classification of the simple graphs whose critical group has two invariant factors equal to one. At the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.4211","kind":"arxiv","version":1},"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":"2026-05-18T02:58:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Rja6KqiMiJ4R+62Lc6nx3OSqItiBlldPi69d3WUV6RiCUjBzQ+HMIf9+J7CNA2nupJY5dNFcKWVom0nQ+EUGBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T12:01:24.410590Z"},"content_sha256":"ccc949b3b65afe6553edcfbfe4295686896056647cfc3d2708c9576fd1b3672b","schema_version":"1.0","event_id":"sha256:ccc949b3b65afe6553edcfbfe4295686896056647cfc3d2708c9576fd1b3672b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KAQBMFMZDM4AMQRYPZOAXY62QB/bundle.json","state_url":"https://pith.science/pith/KAQBMFMZDM4AMQRYPZOAXY62QB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KAQBMFMZDM4AMQRYPZOAXY62QB/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":"2026-06-07T12:01:24Z","links":{"resolver":"https://pith.science/pith/KAQBMFMZDM4AMQRYPZOAXY62QB","bundle":"https://pith.science/pith/KAQBMFMZDM4AMQRYPZOAXY62QB/bundle.json","state":"https://pith.science/pith/KAQBMFMZDM4AMQRYPZOAXY62QB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KAQBMFMZDM4AMQRYPZOAXY62QB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:KAQBMFMZDM4AMQRYPZOAXY62QB","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":"18353431fc6536af6f4b96fb2862f54b7f626f9477b87a5fe379767c8dd6ace1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-04-15T19:26:45Z","title_canon_sha256":"488701670107c2d48cfc0a4fa8e9b15abe2abd7e52bc40c445f8d7dc301acb92"},"schema_version":"1.0","source":{"id":"1304.4211","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.4211","created_at":"2026-05-18T02:58:32Z"},{"alias_kind":"arxiv_version","alias_value":"1304.4211v1","created_at":"2026-05-18T02:58:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.4211","created_at":"2026-05-18T02:58:32Z"},{"alias_kind":"pith_short_12","alias_value":"KAQBMFMZDM4A","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_16","alias_value":"KAQBMFMZDM4AMQRY","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_8","alias_value":"KAQBMFMZ","created_at":"2026-05-18T12:27:49Z"}],"graph_snapshots":[{"event_id":"sha256:ccc949b3b65afe6553edcfbfe4295686896056647cfc3d2708c9576fd1b3672b","target":"graph","created_at":"2026-05-18T02:58:32Z","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 critical ideals of a graph are the determinantal ideals of the generalized Laplacian matrix associated to a graph. A basic property of the critical ideals of graphs asserts that the graphs with at most k trivial critical ideals, $\\Gamma_{\\leq k}$, are closed under induced subgraphs. In this article we find the set of minimal forbidden subgraphs for $\\Gamma_{\\leq 2}$, and we use this forbidden subgraphs to get a classification of the graphs in $\\Gamma_{\\leq 2}$. As a consequence we give a classification of the simple graphs whose critical group has two invariant factors equal to one. At the","authors_text":"Carlos A. Alfaro, Carlos E. Valencia","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-04-15T19:26:45Z","title":"Graphs with two trivial critical ideals"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.4211","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":"sha256:30fb165fa1ec26407475d169161242bd80f1306f5a3a3ab5baab54743e3ee15a","target":"record","created_at":"2026-05-18T02:58:32Z","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":"18353431fc6536af6f4b96fb2862f54b7f626f9477b87a5fe379767c8dd6ace1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-04-15T19:26:45Z","title_canon_sha256":"488701670107c2d48cfc0a4fa8e9b15abe2abd7e52bc40c445f8d7dc301acb92"},"schema_version":"1.0","source":{"id":"1304.4211","kind":"arxiv","version":1}},"canonical_sha256":"50201615991b380642387e5c0be3da8056b1073738afdb230542141b3f551b20","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"50201615991b380642387e5c0be3da8056b1073738afdb230542141b3f551b20","first_computed_at":"2026-05-18T02:58:32.621767Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:58:32.621767Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tdLQrvSZpedN/hln2N38KyAuRvxYzxp0DbZIHxfI/rDSZN3gt8CBISA1XJuN14uVHPwWTuCSeGAWhRxcuoFqDA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:58:32.622399Z","signed_message":"canonical_sha256_bytes"},"source_id":"1304.4211","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:30fb165fa1ec26407475d169161242bd80f1306f5a3a3ab5baab54743e3ee15a","sha256:ccc949b3b65afe6553edcfbfe4295686896056647cfc3d2708c9576fd1b3672b"],"state_sha256":"9ee9ecd8d3a1ea632a1f0e697eefa6f357886e34a152a37f5d0532304e73c8ab"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"b4xd+K5O5tzt1FLhzF/j50Bqy98QVlHjz2eSrf0R4NnoWp5DaSRjNUC2D9SnrIGpTCAmF2e6iKZqdACHt5dwDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T12:01:24.414348Z","bundle_sha256":"1f35da8d6cc330351239282237d360ae3e6de27e3d3c43b55ae536d36bbe7a59"}}