{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:JROGOWEXMOOOL74QGCBVBEKTFO","short_pith_number":"pith:JROGOWEX","canonical_record":{"source":{"id":"1512.02207","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-12-07T20:49:14Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"f578b189006d9348f106565b977481d66e9565f51c85d1d32b824aa62bef2716","abstract_canon_sha256":"64ae9c9d37dd3bed059429cf758e749382517d5bd223e61abeb59c448064daee"},"schema_version":"1.0"},"canonical_sha256":"4c5c675897639ce5ff9030835091532b8cd2260117e0947343cc47c2e7c0f747","source":{"kind":"arxiv","id":"1512.02207","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.02207","created_at":"2026-05-18T01:25:09Z"},{"alias_kind":"arxiv_version","alias_value":"1512.02207v1","created_at":"2026-05-18T01:25:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.02207","created_at":"2026-05-18T01:25:09Z"},{"alias_kind":"pith_short_12","alias_value":"JROGOWEXMOOO","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JROGOWEXMOOOL74Q","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JROGOWEX","created_at":"2026-05-18T12:29:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:JROGOWEXMOOOL74QGCBVBEKTFO","target":"record","payload":{"canonical_record":{"source":{"id":"1512.02207","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-12-07T20:49:14Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"f578b189006d9348f106565b977481d66e9565f51c85d1d32b824aa62bef2716","abstract_canon_sha256":"64ae9c9d37dd3bed059429cf758e749382517d5bd223e61abeb59c448064daee"},"schema_version":"1.0"},"canonical_sha256":"4c5c675897639ce5ff9030835091532b8cd2260117e0947343cc47c2e7c0f747","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:25:09.082033Z","signature_b64":"u+iZT5hZTrX3ac3tlNRuS8Rw4Y55OFj1W+Uws5kkf2/MVsGF131FLxt22kTwA3CL8IFvdiURs2eCyjiN9ZVeBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4c5c675897639ce5ff9030835091532b8cd2260117e0947343cc47c2e7c0f747","last_reissued_at":"2026-05-18T01:25:09.081450Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:25:09.081450Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1512.02207","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":"2026-05-18T01:25:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lWegDLHBi+FWcJlAL3/gRXe1cOU2R8NpwkzQouf9Kbkuh6Axn591++0wcu5yHxesy5E00WeLr2HZMK+nQHrnDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T21:09:53.125184Z"},"content_sha256":"a42615a3c9c5d6a319c0ebfd40356bbd3fe323d2448c6ae14d5271563a77d58b","schema_version":"1.0","event_id":"sha256:a42615a3c9c5d6a319c0ebfd40356bbd3fe323d2448c6ae14d5271563a77d58b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:JROGOWEXMOOOL74QGCBVBEKTFO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The complexity of partitioning into disjoint cliques and a triangle-free graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Marin Bougeret, Pascal Ochem","submitted_at":"2015-12-07T20:49:14Z","abstract_excerpt":"Motivated by Chudnovsky's structure theorem of bull-free graphs, Abu-Khzam, Feghali, and M\\\"uller have recently proved that deciding if a graph has a vertex partition into disjoint cliques and a triangle-free graph is NP-complete for five graph classes. The problem is trivial for the intersection of these five classes. We prove that the problem is NP-complete for the intersection of two subsets of size four among the five classes. We also show NP-completeness for other small classes, such as graphs with maximum degree 4 and line graphs."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.02207","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":"2026-05-18T01:25:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fh5o+X5AktHWUPCBawGUogAo+eVX7hdXWOfqjnkdTyPdeB9KFSrsr3QwiVXhWR9AETYNbrsIhHE3FYY84LdHBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T21:09:53.125901Z"},"content_sha256":"6e1e87cd593f590a8c38f9871dc232cf9e5f0e6f36e9bda4710f2bbfd5146158","schema_version":"1.0","event_id":"sha256:6e1e87cd593f590a8c38f9871dc232cf9e5f0e6f36e9bda4710f2bbfd5146158"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JROGOWEXMOOOL74QGCBVBEKTFO/bundle.json","state_url":"https://pith.science/pith/JROGOWEXMOOOL74QGCBVBEKTFO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JROGOWEXMOOOL74QGCBVBEKTFO/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":"2026-05-27T21:09:53Z","links":{"resolver":"https://pith.science/pith/JROGOWEXMOOOL74QGCBVBEKTFO","bundle":"https://pith.science/pith/JROGOWEXMOOOL74QGCBVBEKTFO/bundle.json","state":"https://pith.science/pith/JROGOWEXMOOOL74QGCBVBEKTFO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JROGOWEXMOOOL74QGCBVBEKTFO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:JROGOWEXMOOOL74QGCBVBEKTFO","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":"64ae9c9d37dd3bed059429cf758e749382517d5bd223e61abeb59c448064daee","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-12-07T20:49:14Z","title_canon_sha256":"f578b189006d9348f106565b977481d66e9565f51c85d1d32b824aa62bef2716"},"schema_version":"1.0","source":{"id":"1512.02207","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.02207","created_at":"2026-05-18T01:25:09Z"},{"alias_kind":"arxiv_version","alias_value":"1512.02207v1","created_at":"2026-05-18T01:25:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.02207","created_at":"2026-05-18T01:25:09Z"},{"alias_kind":"pith_short_12","alias_value":"JROGOWEXMOOO","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JROGOWEXMOOOL74Q","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JROGOWEX","created_at":"2026-05-18T12:29:27Z"}],"graph_snapshots":[{"event_id":"sha256:6e1e87cd593f590a8c38f9871dc232cf9e5f0e6f36e9bda4710f2bbfd5146158","target":"graph","created_at":"2026-05-18T01:25:09Z","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":"Motivated by Chudnovsky's structure theorem of bull-free graphs, Abu-Khzam, Feghali, and M\\\"uller have recently proved that deciding if a graph has a vertex partition into disjoint cliques and a triangle-free graph is NP-complete for five graph classes. The problem is trivial for the intersection of these five classes. We prove that the problem is NP-complete for the intersection of two subsets of size four among the five classes. We also show NP-completeness for other small classes, such as graphs with maximum degree 4 and line graphs.","authors_text":"Marin Bougeret, Pascal Ochem","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-12-07T20:49:14Z","title":"The complexity of partitioning into disjoint cliques and a triangle-free graph"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.02207","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:a42615a3c9c5d6a319c0ebfd40356bbd3fe323d2448c6ae14d5271563a77d58b","target":"record","created_at":"2026-05-18T01:25:09Z","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":"64ae9c9d37dd3bed059429cf758e749382517d5bd223e61abeb59c448064daee","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-12-07T20:49:14Z","title_canon_sha256":"f578b189006d9348f106565b977481d66e9565f51c85d1d32b824aa62bef2716"},"schema_version":"1.0","source":{"id":"1512.02207","kind":"arxiv","version":1}},"canonical_sha256":"4c5c675897639ce5ff9030835091532b8cd2260117e0947343cc47c2e7c0f747","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4c5c675897639ce5ff9030835091532b8cd2260117e0947343cc47c2e7c0f747","first_computed_at":"2026-05-18T01:25:09.081450Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:25:09.081450Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"u+iZT5hZTrX3ac3tlNRuS8Rw4Y55OFj1W+Uws5kkf2/MVsGF131FLxt22kTwA3CL8IFvdiURs2eCyjiN9ZVeBA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:25:09.082033Z","signed_message":"canonical_sha256_bytes"},"source_id":"1512.02207","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a42615a3c9c5d6a319c0ebfd40356bbd3fe323d2448c6ae14d5271563a77d58b","sha256:6e1e87cd593f590a8c38f9871dc232cf9e5f0e6f36e9bda4710f2bbfd5146158"],"state_sha256":"7f23cc204601ae4789f4282946258ead022c8d4aec4049f8d66e723e29027305"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Hw60fk/VwlfwkXlpIlgl+yDo++eIMw2Soi3hEIQZ/U3rsWYfdoxlGlsCnq4f6U6vg8SRhnVEzI+iGxIiTV2YDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T21:09:53.129963Z","bundle_sha256":"1a295223f94de55096347bb466b04278422daf90413b3023da51022acba901ad"}}