{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:I5HH2HYLDFGKEBUIQAT3VWCLGA","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":"1680ea07a0cc7253065cc133a2aa20d41564db8a2f9ea087fbe2d34349f912e5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-03-05T09:12:07Z","title_canon_sha256":"c0ade29c4cb2f898f078bda8717e6570e484d77e9c2e6c45435a09eb16525095"},"schema_version":"1.0","source":{"id":"1103.1028","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1103.1028","created_at":"2026-05-18T04:08:33Z"},{"alias_kind":"arxiv_version","alias_value":"1103.1028v2","created_at":"2026-05-18T04:08:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.1028","created_at":"2026-05-18T04:08:33Z"},{"alias_kind":"pith_short_12","alias_value":"I5HH2HYLDFGK","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_16","alias_value":"I5HH2HYLDFGKEBUI","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_8","alias_value":"I5HH2HYL","created_at":"2026-05-18T12:26:30Z"}],"graph_snapshots":[{"event_id":"sha256:f75214bc8c6d4eafb3ca363b17095e4470239fe784f81e824725a0f4e70afa00","target":"graph","created_at":"2026-05-18T04:08:33Z","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":"The competition graph of a digraph D is a (simple undirected) graph which has the same vertex set as D and has an edge between x and y if and only if there exists a vertex v in D such that (x,v) and (y,v) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of G is the smallest number of such isolated vertices. In general, it is hard to compute the competition number k(G) for a graph G and it has been one of important research problems in the study of competition graphs to characterize ","authors_text":"Boram Park, Jung Yeun Lee, Suh-Ryung Kim, Yoshio Sano","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-03-05T09:12:07Z","title":"The competition number of a graph and the dimension of its hole space"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.1028","kind":"arxiv","version":2},"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":"sha256:35ad145a56878428b636c608c8d0d41c28465c36c3737729c771a9d7a5aa3b3f","target":"record","created_at":"2026-05-18T04:08:33Z","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":"1680ea07a0cc7253065cc133a2aa20d41564db8a2f9ea087fbe2d34349f912e5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-03-05T09:12:07Z","title_canon_sha256":"c0ade29c4cb2f898f078bda8717e6570e484d77e9c2e6c45435a09eb16525095"},"schema_version":"1.0","source":{"id":"1103.1028","kind":"arxiv","version":2}},"canonical_sha256":"474e7d1f0b194ca206888027bad84b30061cd65af5a840028b7d462daa2776ac","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"474e7d1f0b194ca206888027bad84b30061cd65af5a840028b7d462daa2776ac","first_computed_at":"2026-05-18T04:08:33.551364Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:08:33.551364Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9KGnB77TEuNjWvqFloUmhpMfk2AcpuGHGRzlpD2OZJnN9SAIQAjL9ozwOrivIMNh6P6nDHGIwGFxv69gm0juAA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:08:33.551889Z","signed_message":"canonical_sha256_bytes"},"source_id":"1103.1028","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:35ad145a56878428b636c608c8d0d41c28465c36c3737729c771a9d7a5aa3b3f","sha256:f75214bc8c6d4eafb3ca363b17095e4470239fe784f81e824725a0f4e70afa00"],"state_sha256":"ece9842c26b5c5f87de6f5259ecf74a63d85c8178d6c8b91c71f161f8db4aa0b"}