{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:MTCSXTJKAKPR2RQQHEU5FHALIW","short_pith_number":"pith:MTCSXTJK","canonical_record":{"source":{"id":"1612.05827","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2016-12-17T21:32:44Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"ac87a6e98cae7f7262aae9bd457fba57d5c9df884a27dcc6b7c93dd7b24ff2d8","abstract_canon_sha256":"02bb1c722de33c725961a56cc0c7d090f2e3336e6c16a13c29f8fdaa47e0c755"},"schema_version":"1.0"},"canonical_sha256":"64c52bcd2a029f1d46103929d29c0b458125e78bfe92b6b1663b96a7fa12a6a6","source":{"kind":"arxiv","id":"1612.05827","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.05827","created_at":"2026-05-18T00:02:03Z"},{"alias_kind":"arxiv_version","alias_value":"1612.05827v1","created_at":"2026-05-18T00:02:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.05827","created_at":"2026-05-18T00:02:03Z"},{"alias_kind":"pith_short_12","alias_value":"MTCSXTJKAKPR","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_16","alias_value":"MTCSXTJKAKPR2RQQ","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_8","alias_value":"MTCSXTJK","created_at":"2026-05-18T12:30:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:MTCSXTJKAKPR2RQQHEU5FHALIW","target":"record","payload":{"canonical_record":{"source":{"id":"1612.05827","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2016-12-17T21:32:44Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"ac87a6e98cae7f7262aae9bd457fba57d5c9df884a27dcc6b7c93dd7b24ff2d8","abstract_canon_sha256":"02bb1c722de33c725961a56cc0c7d090f2e3336e6c16a13c29f8fdaa47e0c755"},"schema_version":"1.0"},"canonical_sha256":"64c52bcd2a029f1d46103929d29c0b458125e78bfe92b6b1663b96a7fa12a6a6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:02:03.394392Z","signature_b64":"fp4xGYx1hAd1ZTJpaZZ5wnC/RZZoLdnRUB2j026y1jzkMe9jjTQlvAjc6SZCIx1/D9QI3J/ys45Jydha5XsdAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"64c52bcd2a029f1d46103929d29c0b458125e78bfe92b6b1663b96a7fa12a6a6","last_reissued_at":"2026-05-18T00:02:03.393885Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:02:03.393885Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1612.05827","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":"2026-05-18T00:02:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mXwFAA/x68aTYlhHOb14iF70ybuF0YQbeetpDV9JutGKrHzUKFZdmn8nrW9ZPogZ1Z57caTdUyXF4bQEEJ9fBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T06:59:56.364378Z"},"content_sha256":"e3dfdc2bc97e330a19a10cc75ff6821d7bef9446039fb31424d3ba732e409f13","schema_version":"1.0","event_id":"sha256:e3dfdc2bc97e330a19a10cc75ff6821d7bef9446039fb31424d3ba732e409f13"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:MTCSXTJKAKPR2RQQHEU5FHALIW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Cograph generation with linear delay","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"\\'Atila A. Jones, F\\'abio Protti, Renata R. Del-Vecchio","submitted_at":"2016-12-17T21:32:44Z","abstract_excerpt":"Cographs have always been a research target in areas such as coloring, graph decomposition, and spectral theory. In this work, we present an algorithm to generate all unlabeled cographs with $n$ vertices, based on the generation of cotrees. The delay of our algorithm (time spent between two consecutive outputs) is $O(n)$. The time needed to generate the first output is also $O(n)$, which gives an overall $O(n\\,M_n)$ time complexity, where $M_n$ is the number of unlabeled cographs with $n$ vertices. The algorithm avoids the generation of duplicates (isomorphic outputs) and produces, as a by-pro"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.05827","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":"2026-05-18T00:02:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"j5xMIqlG8KHgPumdqQnixWksR7f9yXPQQT3FjDbes4UQr7j4CEJ48+HMXo7oF/kyn+3VZWKsa9s4GENp/9xRAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T06:59:56.365114Z"},"content_sha256":"552b0dfb36a69a97884f575838f2a36933560f4cbdcc477b589ccdeff0f4cd52","schema_version":"1.0","event_id":"sha256:552b0dfb36a69a97884f575838f2a36933560f4cbdcc477b589ccdeff0f4cd52"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MTCSXTJKAKPR2RQQHEU5FHALIW/bundle.json","state_url":"https://pith.science/pith/MTCSXTJKAKPR2RQQHEU5FHALIW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MTCSXTJKAKPR2RQQHEU5FHALIW/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":"2026-06-11T06:59:56Z","links":{"resolver":"https://pith.science/pith/MTCSXTJKAKPR2RQQHEU5FHALIW","bundle":"https://pith.science/pith/MTCSXTJKAKPR2RQQHEU5FHALIW/bundle.json","state":"https://pith.science/pith/MTCSXTJKAKPR2RQQHEU5FHALIW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MTCSXTJKAKPR2RQQHEU5FHALIW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:MTCSXTJKAKPR2RQQHEU5FHALIW","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":"02bb1c722de33c725961a56cc0c7d090f2e3336e6c16a13c29f8fdaa47e0c755","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2016-12-17T21:32:44Z","title_canon_sha256":"ac87a6e98cae7f7262aae9bd457fba57d5c9df884a27dcc6b7c93dd7b24ff2d8"},"schema_version":"1.0","source":{"id":"1612.05827","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.05827","created_at":"2026-05-18T00:02:03Z"},{"alias_kind":"arxiv_version","alias_value":"1612.05827v1","created_at":"2026-05-18T00:02:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.05827","created_at":"2026-05-18T00:02:03Z"},{"alias_kind":"pith_short_12","alias_value":"MTCSXTJKAKPR","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_16","alias_value":"MTCSXTJKAKPR2RQQ","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_8","alias_value":"MTCSXTJK","created_at":"2026-05-18T12:30:32Z"}],"graph_snapshots":[{"event_id":"sha256:552b0dfb36a69a97884f575838f2a36933560f4cbdcc477b589ccdeff0f4cd52","target":"graph","created_at":"2026-05-18T00:02:03Z","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":"Cographs have always been a research target in areas such as coloring, graph decomposition, and spectral theory. In this work, we present an algorithm to generate all unlabeled cographs with $n$ vertices, based on the generation of cotrees. The delay of our algorithm (time spent between two consecutive outputs) is $O(n)$. The time needed to generate the first output is also $O(n)$, which gives an overall $O(n\\,M_n)$ time complexity, where $M_n$ is the number of unlabeled cographs with $n$ vertices. The algorithm avoids the generation of duplicates (isomorphic outputs) and produces, as a by-pro","authors_text":"\\'Atila A. Jones, F\\'abio Protti, Renata R. Del-Vecchio","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2016-12-17T21:32:44Z","title":"Cograph generation with linear delay"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.05827","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:e3dfdc2bc97e330a19a10cc75ff6821d7bef9446039fb31424d3ba732e409f13","target":"record","created_at":"2026-05-18T00:02:03Z","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":"02bb1c722de33c725961a56cc0c7d090f2e3336e6c16a13c29f8fdaa47e0c755","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2016-12-17T21:32:44Z","title_canon_sha256":"ac87a6e98cae7f7262aae9bd457fba57d5c9df884a27dcc6b7c93dd7b24ff2d8"},"schema_version":"1.0","source":{"id":"1612.05827","kind":"arxiv","version":1}},"canonical_sha256":"64c52bcd2a029f1d46103929d29c0b458125e78bfe92b6b1663b96a7fa12a6a6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"64c52bcd2a029f1d46103929d29c0b458125e78bfe92b6b1663b96a7fa12a6a6","first_computed_at":"2026-05-18T00:02:03.393885Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:02:03.393885Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fp4xGYx1hAd1ZTJpaZZ5wnC/RZZoLdnRUB2j026y1jzkMe9jjTQlvAjc6SZCIx1/D9QI3J/ys45Jydha5XsdAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:02:03.394392Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.05827","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e3dfdc2bc97e330a19a10cc75ff6821d7bef9446039fb31424d3ba732e409f13","sha256:552b0dfb36a69a97884f575838f2a36933560f4cbdcc477b589ccdeff0f4cd52"],"state_sha256":"9d4202e3dd44173911c7e974245d0476f15013c348ceb68781ea6c26961b0d6e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"V0uC2mu1Fw/QukPKHsQoiP/AmzuQrM4xEP3tY06+g7VS/voTWFYW2g9Xtmwin8/KdDkGM7ce05VkkR8tDYEpDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T06:59:56.369854Z","bundle_sha256":"478de9cde679912d7c978e05f7a5874c43c4cd4a9b20fc9a87cec7b4e9d05a2b"}}