{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:KGMX3XILLNNPXA6VVVAL3IBKM5","short_pith_number":"pith:KGMX3XIL","schema_version":"1.0","canonical_sha256":"51997ddd0b5b5afb83d5ad40bda02a6748e98bb35b742f8a2593d135502cb5a6","source":{"kind":"arxiv","id":"1410.4084","version":1},"attestation_state":"computed","paper":{"title":"Implicit Representations and Factorial Properties of Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aistis Atminas, Andrew Collins, Vadim Lozin, Victor Zamaraev","submitted_at":"2014-10-15T14:55:52Z","abstract_excerpt":"The idea of implicit representation of graphs was introduced in [S. Kannan, M. Naor, S. Rudich, Implicit representation of graphs, SIAM J. Discrete Mathematics, 5 (1992) 596--603] and can be defined as follows. A representation of an $n$-vertex graph $G$ is said to be implicit if it assigns to each vertex of $G$ a binary code of length $O(\\log n)$ so that the adjacency of two vertices is a function of their codes. Since an implicit representation of an $n$-vertex graph uses $O(n\\log n)$ bits, any class of graphs admitting such a representation contains $2^{O(n\\log n)}$ labelled graphs with $n$"},"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":"1410.4084","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-10-15T14:55:52Z","cross_cats_sorted":[],"title_canon_sha256":"d6d22fb4da0f6766c5f596d8d69a0593dec3495842ff393fd522178385dfff0e","abstract_canon_sha256":"b852e5f5e6b4000c56cbcf967a0d4935294a507df5ed2f5835146af251eb93d9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:40:02.671240Z","signature_b64":"lghdDHclp8j6u2fT2jlBqw65GUadkValucNbuwwN5CgI69kyljhKBuOQUg/joMUJtXzMyRqehgApJL4BUy1HAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"51997ddd0b5b5afb83d5ad40bda02a6748e98bb35b742f8a2593d135502cb5a6","last_reissued_at":"2026-05-18T02:40:02.670781Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:40:02.670781Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Implicit Representations and Factorial Properties of Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aistis Atminas, Andrew Collins, Vadim Lozin, Victor Zamaraev","submitted_at":"2014-10-15T14:55:52Z","abstract_excerpt":"The idea of implicit representation of graphs was introduced in [S. Kannan, M. Naor, S. Rudich, Implicit representation of graphs, SIAM J. Discrete Mathematics, 5 (1992) 596--603] and can be defined as follows. A representation of an $n$-vertex graph $G$ is said to be implicit if it assigns to each vertex of $G$ a binary code of length $O(\\log n)$ so that the adjacency of two vertices is a function of their codes. Since an implicit representation of an $n$-vertex graph uses $O(n\\log n)$ bits, any class of graphs admitting such a representation contains $2^{O(n\\log n)}$ labelled graphs with $n$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.4084","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1410.4084","created_at":"2026-05-18T02:40:02.670851+00:00"},{"alias_kind":"arxiv_version","alias_value":"1410.4084v1","created_at":"2026-05-18T02:40:02.670851+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.4084","created_at":"2026-05-18T02:40:02.670851+00:00"},{"alias_kind":"pith_short_12","alias_value":"KGMX3XILLNNP","created_at":"2026-05-18T12:28:35.611951+00:00"},{"alias_kind":"pith_short_16","alias_value":"KGMX3XILLNNPXA6V","created_at":"2026-05-18T12:28:35.611951+00:00"},{"alias_kind":"pith_short_8","alias_value":"KGMX3XIL","created_at":"2026-05-18T12:28:35.611951+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/KGMX3XILLNNPXA6VVVAL3IBKM5","json":"https://pith.science/pith/KGMX3XILLNNPXA6VVVAL3IBKM5.json","graph_json":"https://pith.science/api/pith-number/KGMX3XILLNNPXA6VVVAL3IBKM5/graph.json","events_json":"https://pith.science/api/pith-number/KGMX3XILLNNPXA6VVVAL3IBKM5/events.json","paper":"https://pith.science/paper/KGMX3XIL"},"agent_actions":{"view_html":"https://pith.science/pith/KGMX3XILLNNPXA6VVVAL3IBKM5","download_json":"https://pith.science/pith/KGMX3XILLNNPXA6VVVAL3IBKM5.json","view_paper":"https://pith.science/paper/KGMX3XIL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1410.4084&json=true","fetch_graph":"https://pith.science/api/pith-number/KGMX3XILLNNPXA6VVVAL3IBKM5/graph.json","fetch_events":"https://pith.science/api/pith-number/KGMX3XILLNNPXA6VVVAL3IBKM5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KGMX3XILLNNPXA6VVVAL3IBKM5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KGMX3XILLNNPXA6VVVAL3IBKM5/action/storage_attestation","attest_author":"https://pith.science/pith/KGMX3XILLNNPXA6VVVAL3IBKM5/action/author_attestation","sign_citation":"https://pith.science/pith/KGMX3XILLNNPXA6VVVAL3IBKM5/action/citation_signature","submit_replication":"https://pith.science/pith/KGMX3XILLNNPXA6VVVAL3IBKM5/action/replication_record"}},"created_at":"2026-05-18T02:40:02.670851+00:00","updated_at":"2026-05-18T02:40:02.670851+00:00"}