{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2022:J2GKJGOOINM3VBTOIMET4FPE5I","short_pith_number":"pith:J2GKJGOO","canonical_record":{"source":{"id":"2201.00721","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AT","submitted_at":"2022-01-03T15:48:40Z","cross_cats_sorted":["math.CO","math.GT"],"title_canon_sha256":"c26cf5cd1762d8f921c5a9884f0b5f61e020508cd9f31aa58be9dcd069b3e4f8","abstract_canon_sha256":"88292ee78b6ae70e69788d1b612ced0dd78394c0aa59fca6596592717b5b4800"},"schema_version":"1.0"},"canonical_sha256":"4e8ca499ce4359ba866e43093e15e4ea2e16276a8b938da0c63e669e4c8f72d1","source":{"kind":"arxiv","id":"2201.00721","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2201.00721","created_at":"2026-07-05T06:41:43Z"},{"alias_kind":"arxiv_version","alias_value":"2201.00721v1","created_at":"2026-07-05T06:41:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2201.00721","created_at":"2026-07-05T06:41:43Z"},{"alias_kind":"pith_short_12","alias_value":"J2GKJGOOINM3","created_at":"2026-07-05T06:41:43Z"},{"alias_kind":"pith_short_16","alias_value":"J2GKJGOOINM3VBTO","created_at":"2026-07-05T06:41:43Z"},{"alias_kind":"pith_short_8","alias_value":"J2GKJGOO","created_at":"2026-07-05T06:41:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2022:J2GKJGOOINM3VBTOIMET4FPE5I","target":"record","payload":{"canonical_record":{"source":{"id":"2201.00721","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AT","submitted_at":"2022-01-03T15:48:40Z","cross_cats_sorted":["math.CO","math.GT"],"title_canon_sha256":"c26cf5cd1762d8f921c5a9884f0b5f61e020508cd9f31aa58be9dcd069b3e4f8","abstract_canon_sha256":"88292ee78b6ae70e69788d1b612ced0dd78394c0aa59fca6596592717b5b4800"},"schema_version":"1.0"},"canonical_sha256":"4e8ca499ce4359ba866e43093e15e4ea2e16276a8b938da0c63e669e4c8f72d1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:41:43.449269Z","signature_b64":"5yJgKAsMZulrhEiAoQbTd0qKmG8goccsiBtrWuiUMZxbYcFoGRh0MyPpar6VbK8Mibi8vHoSiJwq69a96/cHAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4e8ca499ce4359ba866e43093e15e4ea2e16276a8b938da0c63e669e4c8f72d1","last_reissued_at":"2026-07-05T06:41:43.448745Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:41:43.448745Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2201.00721","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":"2026-07-05T06:41:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NzeLMm1NJVfHK7QZywbeA2WDZCKUNYKZJA/uKc4X6VxY/aoWoRYPOMVwsEnun/jiKnrgcmwoZIuvNBf3Qp+MAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T10:02:08.667859Z"},"content_sha256":"338d142c01dbf3d00dee0f5ebc89cdd3228dd71717ad4d67c6f044cd3c818b1a","schema_version":"1.0","event_id":"sha256:338d142c01dbf3d00dee0f5ebc89cdd3228dd71717ad4d67c6f044cd3c818b1a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2022:J2GKJGOOINM3VBTOIMET4FPE5I","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Categorifying connected domination via graph \\\"uberhomology","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["math.CO","math.GT"],"primary_cat":"math.AT","authors_text":"Carlo Collari, Daniele Celoria, Luigi Caputi","submitted_at":"2022-01-03T15:48:40Z","abstract_excerpt":"\\\"Uberhomology is a recently defined homology theory for simplicial complexes, which yields subtle information on graphs. We prove that bold homology, a certain specialisation of \\\"uberhomology, is related to dominating sets in graphs. To this end, we interpret \\\"uberhomology as a poset homology, and investigate its functoriality properties. We then show that the Euler characteristic of the bold homology of a graph coincides with an evaluation of its connected domination polynomial. Even more, the bold chain complex retracts onto a complex generated by connected dominating sets. We conclude wi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2201.00721","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2201.00721/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T06:41:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eYxsfkWzoT+um4X/X2/LW6nUkyyB2NPiMpkciygBQLl2Pj/tkdSYFnrfDoQOpvld+DB8xwLY5LQ1JvD5nB8tCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T10:02:08.668835Z"},"content_sha256":"9acc546dd71e3dd83e53379bf4518e8d32b4f76eee02ca5f5d55ff66fa0e23ba","schema_version":"1.0","event_id":"sha256:9acc546dd71e3dd83e53379bf4518e8d32b4f76eee02ca5f5d55ff66fa0e23ba"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/J2GKJGOOINM3VBTOIMET4FPE5I/bundle.json","state_url":"https://pith.science/pith/J2GKJGOOINM3VBTOIMET4FPE5I/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/J2GKJGOOINM3VBTOIMET4FPE5I/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":"2026-08-04T10:02:08Z","links":{"resolver":"https://pith.science/pith/J2GKJGOOINM3VBTOIMET4FPE5I","bundle":"https://pith.science/pith/J2GKJGOOINM3VBTOIMET4FPE5I/bundle.json","state":"https://pith.science/pith/J2GKJGOOINM3VBTOIMET4FPE5I/state.json","well_known_bundle":"https://pith.science/.well-known/pith/J2GKJGOOINM3VBTOIMET4FPE5I/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:J2GKJGOOINM3VBTOIMET4FPE5I","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":"88292ee78b6ae70e69788d1b612ced0dd78394c0aa59fca6596592717b5b4800","cross_cats_sorted":["math.CO","math.GT"],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AT","submitted_at":"2022-01-03T15:48:40Z","title_canon_sha256":"c26cf5cd1762d8f921c5a9884f0b5f61e020508cd9f31aa58be9dcd069b3e4f8"},"schema_version":"1.0","source":{"id":"2201.00721","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2201.00721","created_at":"2026-07-05T06:41:43Z"},{"alias_kind":"arxiv_version","alias_value":"2201.00721v1","created_at":"2026-07-05T06:41:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2201.00721","created_at":"2026-07-05T06:41:43Z"},{"alias_kind":"pith_short_12","alias_value":"J2GKJGOOINM3","created_at":"2026-07-05T06:41:43Z"},{"alias_kind":"pith_short_16","alias_value":"J2GKJGOOINM3VBTO","created_at":"2026-07-05T06:41:43Z"},{"alias_kind":"pith_short_8","alias_value":"J2GKJGOO","created_at":"2026-07-05T06:41:43Z"}],"graph_snapshots":[{"event_id":"sha256:9acc546dd71e3dd83e53379bf4518e8d32b4f76eee02ca5f5d55ff66fa0e23ba","target":"graph","created_at":"2026-07-05T06:41:43Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2201.00721/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"\\\"Uberhomology is a recently defined homology theory for simplicial complexes, which yields subtle information on graphs. We prove that bold homology, a certain specialisation of \\\"uberhomology, is related to dominating sets in graphs. To this end, we interpret \\\"uberhomology as a poset homology, and investigate its functoriality properties. We then show that the Euler characteristic of the bold homology of a graph coincides with an evaluation of its connected domination polynomial. Even more, the bold chain complex retracts onto a complex generated by connected dominating sets. We conclude wi","authors_text":"Carlo Collari, Daniele Celoria, Luigi Caputi","cross_cats":["math.CO","math.GT"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AT","submitted_at":"2022-01-03T15:48:40Z","title":"Categorifying connected domination via graph \\\"uberhomology"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2201.00721","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":"sha256:338d142c01dbf3d00dee0f5ebc89cdd3228dd71717ad4d67c6f044cd3c818b1a","target":"record","created_at":"2026-07-05T06:41:43Z","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":"88292ee78b6ae70e69788d1b612ced0dd78394c0aa59fca6596592717b5b4800","cross_cats_sorted":["math.CO","math.GT"],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AT","submitted_at":"2022-01-03T15:48:40Z","title_canon_sha256":"c26cf5cd1762d8f921c5a9884f0b5f61e020508cd9f31aa58be9dcd069b3e4f8"},"schema_version":"1.0","source":{"id":"2201.00721","kind":"arxiv","version":1}},"canonical_sha256":"4e8ca499ce4359ba866e43093e15e4ea2e16276a8b938da0c63e669e4c8f72d1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4e8ca499ce4359ba866e43093e15e4ea2e16276a8b938da0c63e669e4c8f72d1","first_computed_at":"2026-07-05T06:41:43.448745Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T06:41:43.448745Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5yJgKAsMZulrhEiAoQbTd0qKmG8goccsiBtrWuiUMZxbYcFoGRh0MyPpar6VbK8Mibi8vHoSiJwq69a96/cHAg==","signature_status":"signed_v1","signed_at":"2026-07-05T06:41:43.449269Z","signed_message":"canonical_sha256_bytes"},"source_id":"2201.00721","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:338d142c01dbf3d00dee0f5ebc89cdd3228dd71717ad4d67c6f044cd3c818b1a","sha256:9acc546dd71e3dd83e53379bf4518e8d32b4f76eee02ca5f5d55ff66fa0e23ba"],"state_sha256":"aa0041d332ca233bd4f9a99de1a60185f69f542fa5e452459a7e8b8284ebff07"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UgqsuXWzA6OoWzTnmdnFXgsX9c5mptJsNV/LPbqKkxFt9JeG4BsePE6NP8VJR9JmDHinqTPQcGoSQtvHEMkiCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-04T10:02:08.675805Z","bundle_sha256":"eedff93ea44b7addc88172754501401ee5afbe44bf93a608f6755ce7c2b79214"}}