{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:X73C7BEPYLODAO7UOOWLATMZBU","short_pith_number":"pith:X73C7BEP","schema_version":"1.0","canonical_sha256":"bff62f848fc2dc303bf473acb04d990d3e71d8117be1039d503003835a04b087","source":{"kind":"arxiv","id":"1011.4726","version":2},"attestation_state":"computed","paper":{"title":"$H$-product and $H$-threshold graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Pavel Skums","submitted_at":"2010-11-22T04:43:20Z","abstract_excerpt":"This paper is the continuation of the research of the author and his colleagues of the {\\it canonical} decomposition of graphs. The idea of the canonical decomposition is to define the binary operation on the set of graphs and to represent the graph under study as a product of prime elements with respect to this operation. We consider the graph together with the arbitrary partition of its vertex set into $n$ subsets ($n$-partitioned graph). On the set of $n$-partitioned graphs distinguished up to isomorphism we consider the binary algebraic operation $\\circ_H$ ($H$-product of graphs), determin"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1011.4726","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-11-22T04:43:20Z","cross_cats_sorted":[],"title_canon_sha256":"bc8c51ed93659db3f214bcb81f2ab364ffa62ebf71dda35da91632bfe1ed968a","abstract_canon_sha256":"bc030b123fcbe71baf4d2842b9dcc0be79d9c391fec77b9cfd926ea1325d0625"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:11:00.695469Z","signature_b64":"TSVML5wfceY6wBkS0faIHTlDukwPK7SW+iabG2WjHcsjexHML4qrKCVgdtfilCn+Ez11sjuwcMQttnETKV+cCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bff62f848fc2dc303bf473acb04d990d3e71d8117be1039d503003835a04b087","last_reissued_at":"2026-05-18T01:11:00.694837Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:11:00.694837Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$H$-product and $H$-threshold graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Pavel Skums","submitted_at":"2010-11-22T04:43:20Z","abstract_excerpt":"This paper is the continuation of the research of the author and his colleagues of the {\\it canonical} decomposition of graphs. The idea of the canonical decomposition is to define the binary operation on the set of graphs and to represent the graph under study as a product of prime elements with respect to this operation. We consider the graph together with the arbitrary partition of its vertex set into $n$ subsets ($n$-partitioned graph). On the set of $n$-partitioned graphs distinguished up to isomorphism we consider the binary algebraic operation $\\circ_H$ ($H$-product of graphs), determin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.4726","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1011.4726","created_at":"2026-05-18T01:11:00.694922+00:00"},{"alias_kind":"arxiv_version","alias_value":"1011.4726v2","created_at":"2026-05-18T01:11:00.694922+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.4726","created_at":"2026-05-18T01:11:00.694922+00:00"},{"alias_kind":"pith_short_12","alias_value":"X73C7BEPYLOD","created_at":"2026-05-18T12:26:17.028572+00:00"},{"alias_kind":"pith_short_16","alias_value":"X73C7BEPYLODAO7U","created_at":"2026-05-18T12:26:17.028572+00:00"},{"alias_kind":"pith_short_8","alias_value":"X73C7BEP","created_at":"2026-05-18T12:26:17.028572+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/X73C7BEPYLODAO7UOOWLATMZBU","json":"https://pith.science/pith/X73C7BEPYLODAO7UOOWLATMZBU.json","graph_json":"https://pith.science/api/pith-number/X73C7BEPYLODAO7UOOWLATMZBU/graph.json","events_json":"https://pith.science/api/pith-number/X73C7BEPYLODAO7UOOWLATMZBU/events.json","paper":"https://pith.science/paper/X73C7BEP"},"agent_actions":{"view_html":"https://pith.science/pith/X73C7BEPYLODAO7UOOWLATMZBU","download_json":"https://pith.science/pith/X73C7BEPYLODAO7UOOWLATMZBU.json","view_paper":"https://pith.science/paper/X73C7BEP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1011.4726&json=true","fetch_graph":"https://pith.science/api/pith-number/X73C7BEPYLODAO7UOOWLATMZBU/graph.json","fetch_events":"https://pith.science/api/pith-number/X73C7BEPYLODAO7UOOWLATMZBU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/X73C7BEPYLODAO7UOOWLATMZBU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/X73C7BEPYLODAO7UOOWLATMZBU/action/storage_attestation","attest_author":"https://pith.science/pith/X73C7BEPYLODAO7UOOWLATMZBU/action/author_attestation","sign_citation":"https://pith.science/pith/X73C7BEPYLODAO7UOOWLATMZBU/action/citation_signature","submit_replication":"https://pith.science/pith/X73C7BEPYLODAO7UOOWLATMZBU/action/replication_record"}},"created_at":"2026-05-18T01:11:00.694922+00:00","updated_at":"2026-05-18T01:11:00.694922+00:00"}