{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:7Q6GTHDDZY4YJPH4XYAMBAUD74","short_pith_number":"pith:7Q6GTHDD","canonical_record":{"source":{"id":"1710.03386","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-10-10T03:21:22Z","cross_cats_sorted":[],"title_canon_sha256":"94b5e358a26f44e05b1d15fbac2dae2c933999e6c4da963d17c6cb6cf1a75c21","abstract_canon_sha256":"ab3aea75bf37b475fc0813111941c9102c8ed11763f556b297d05300819e77e2"},"schema_version":"1.0"},"canonical_sha256":"fc3c699c63ce3984bcfcbe00c08283ff2d667bdc2956f6e552d002b29fc14c56","source":{"kind":"arxiv","id":"1710.03386","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1710.03386","created_at":"2026-05-18T00:33:12Z"},{"alias_kind":"arxiv_version","alias_value":"1710.03386v1","created_at":"2026-05-18T00:33:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.03386","created_at":"2026-05-18T00:33:12Z"},{"alias_kind":"pith_short_12","alias_value":"7Q6GTHDDZY4Y","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_16","alias_value":"7Q6GTHDDZY4YJPH4","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_8","alias_value":"7Q6GTHDD","created_at":"2026-05-18T12:31:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:7Q6GTHDDZY4YJPH4XYAMBAUD74","target":"record","payload":{"canonical_record":{"source":{"id":"1710.03386","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-10-10T03:21:22Z","cross_cats_sorted":[],"title_canon_sha256":"94b5e358a26f44e05b1d15fbac2dae2c933999e6c4da963d17c6cb6cf1a75c21","abstract_canon_sha256":"ab3aea75bf37b475fc0813111941c9102c8ed11763f556b297d05300819e77e2"},"schema_version":"1.0"},"canonical_sha256":"fc3c699c63ce3984bcfcbe00c08283ff2d667bdc2956f6e552d002b29fc14c56","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:33:12.102147Z","signature_b64":"U8zqwZ39Os15yjq4Z0xKvzhMXzYRq3VcQ2hop4r/sKdlm/Bp6yfsgMTGAYFXSi7tDWUEJzDn0ei1536qPlkSDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fc3c699c63ce3984bcfcbe00c08283ff2d667bdc2956f6e552d002b29fc14c56","last_reissued_at":"2026-05-18T00:33:12.101449Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:33:12.101449Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1710.03386","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":"2026-05-18T00:33:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pD9zvD7Z4Z5/LaKUQQi/jTqDJOx8rmadOd3duXsMmMmPcwYZ8lO9o628KNmoCY7w19h6NtJdP6fBgbTNy/JbBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T10:16:54.788676Z"},"content_sha256":"96f848a233802a7fafc396cb2d0bb24651a09f72ec877682e9c9a9b7fae481fc","schema_version":"1.0","event_id":"sha256:96f848a233802a7fafc396cb2d0bb24651a09f72ec877682e9c9a9b7fae481fc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:7Q6GTHDDZY4YJPH4XYAMBAUD74","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Critical ideals, minimum rank and zero forcing number","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Carlos A. Alfaro, Jephian C.-H. Lin","submitted_at":"2017-10-10T03:21:22Z","abstract_excerpt":"There are profound relations between the zero forcing number and minimum rank of a graph. We study the relation of both parameters with a third one, the algebraic co-rank; that is defined as the largest $i$ such that the $i$-th critical ideal is trivial. This gives a new perspective for bounding and computing these three graph parameters."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.03386","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":"2026-05-18T00:33:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Iuuj8KEtrrDS2UGaa/wAntQqwdymKimie0HvDo50j+H7A92BssIyV0T+nRUCMFEnQGxZ1QrCo/JEP3AN0SBDCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T10:16:54.789021Z"},"content_sha256":"c0e3eb203b14a11ce099b0e74f0da7a56b577362a9ece647a22bd32ebe3df92a","schema_version":"1.0","event_id":"sha256:c0e3eb203b14a11ce099b0e74f0da7a56b577362a9ece647a22bd32ebe3df92a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7Q6GTHDDZY4YJPH4XYAMBAUD74/bundle.json","state_url":"https://pith.science/pith/7Q6GTHDDZY4YJPH4XYAMBAUD74/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7Q6GTHDDZY4YJPH4XYAMBAUD74/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":"2026-06-22T10:16:54Z","links":{"resolver":"https://pith.science/pith/7Q6GTHDDZY4YJPH4XYAMBAUD74","bundle":"https://pith.science/pith/7Q6GTHDDZY4YJPH4XYAMBAUD74/bundle.json","state":"https://pith.science/pith/7Q6GTHDDZY4YJPH4XYAMBAUD74/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7Q6GTHDDZY4YJPH4XYAMBAUD74/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:7Q6GTHDDZY4YJPH4XYAMBAUD74","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":"ab3aea75bf37b475fc0813111941c9102c8ed11763f556b297d05300819e77e2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-10-10T03:21:22Z","title_canon_sha256":"94b5e358a26f44e05b1d15fbac2dae2c933999e6c4da963d17c6cb6cf1a75c21"},"schema_version":"1.0","source":{"id":"1710.03386","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1710.03386","created_at":"2026-05-18T00:33:12Z"},{"alias_kind":"arxiv_version","alias_value":"1710.03386v1","created_at":"2026-05-18T00:33:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.03386","created_at":"2026-05-18T00:33:12Z"},{"alias_kind":"pith_short_12","alias_value":"7Q6GTHDDZY4Y","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_16","alias_value":"7Q6GTHDDZY4YJPH4","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_8","alias_value":"7Q6GTHDD","created_at":"2026-05-18T12:31:05Z"}],"graph_snapshots":[{"event_id":"sha256:c0e3eb203b14a11ce099b0e74f0da7a56b577362a9ece647a22bd32ebe3df92a","target":"graph","created_at":"2026-05-18T00:33:12Z","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":"There are profound relations between the zero forcing number and minimum rank of a graph. We study the relation of both parameters with a third one, the algebraic co-rank; that is defined as the largest $i$ such that the $i$-th critical ideal is trivial. This gives a new perspective for bounding and computing these three graph parameters.","authors_text":"Carlos A. Alfaro, Jephian C.-H. Lin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-10-10T03:21:22Z","title":"Critical ideals, minimum rank and zero forcing number"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.03386","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":"sha256:96f848a233802a7fafc396cb2d0bb24651a09f72ec877682e9c9a9b7fae481fc","target":"record","created_at":"2026-05-18T00:33:12Z","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":"ab3aea75bf37b475fc0813111941c9102c8ed11763f556b297d05300819e77e2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-10-10T03:21:22Z","title_canon_sha256":"94b5e358a26f44e05b1d15fbac2dae2c933999e6c4da963d17c6cb6cf1a75c21"},"schema_version":"1.0","source":{"id":"1710.03386","kind":"arxiv","version":1}},"canonical_sha256":"fc3c699c63ce3984bcfcbe00c08283ff2d667bdc2956f6e552d002b29fc14c56","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fc3c699c63ce3984bcfcbe00c08283ff2d667bdc2956f6e552d002b29fc14c56","first_computed_at":"2026-05-18T00:33:12.101449Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:33:12.101449Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"U8zqwZ39Os15yjq4Z0xKvzhMXzYRq3VcQ2hop4r/sKdlm/Bp6yfsgMTGAYFXSi7tDWUEJzDn0ei1536qPlkSDw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:33:12.102147Z","signed_message":"canonical_sha256_bytes"},"source_id":"1710.03386","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:96f848a233802a7fafc396cb2d0bb24651a09f72ec877682e9c9a9b7fae481fc","sha256:c0e3eb203b14a11ce099b0e74f0da7a56b577362a9ece647a22bd32ebe3df92a"],"state_sha256":"bae00589559bed117975dc60bb3e4f4cb7039b6c6c126a893ae1ba10b851c584"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"osP3KPFPEbV1CPyf98NXQCY38enFm4q7VT9TH6UyPnnprx9GJlp07ZAl9mf8z/MbWAuSq+OTN1+TgyeQbhYcAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T10:16:54.790959Z","bundle_sha256":"abcd1338d86f773c7988747356de6170c1a4c09508aa9b6663010856c985605b"}}