{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:LFC65WHSJHKXDUCGB3UYESBSGR","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":"09ca263c4fc9994e11d9ac427d4bcd47078e364341f80b1cff94e8c945b45f70","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-11-15T12:02:28Z","title_canon_sha256":"6b4ba5f087a36609c7b2c370f27143f213daf56cb91fe124293cd6eeae69bb7b"},"schema_version":"1.0","source":{"id":"1111.3512","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.3512","created_at":"2026-05-18T03:58:39Z"},{"alias_kind":"arxiv_version","alias_value":"1111.3512v2","created_at":"2026-05-18T03:58:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.3512","created_at":"2026-05-18T03:58:39Z"},{"alias_kind":"pith_short_12","alias_value":"LFC65WHSJHKX","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_16","alias_value":"LFC65WHSJHKXDUCG","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_8","alias_value":"LFC65WHS","created_at":"2026-05-18T12:26:34Z"}],"graph_snapshots":[{"event_id":"sha256:f3e0ab72c5aeb77454f98ad0d25d2afabe7964b95eeb98b4584f92e4e52177f8","target":"graph","created_at":"2026-05-18T03:58:39Z","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":"A set of vertices $S$ of a graph $G$ is a geodetic set of $G$ if every vertex $v\\not\\in S$ lies on a shortest path between two vertices of $S$. The minimum cardinality of a geodetic set of $G$ is the geodetic number of $G$ and it is denoted by $g(G)$. A Steiner set of $G$ is a set of vertices $W$ of $G$ such that every vertex of $G$ belongs to the set of vertices of a connected subgraph of minimum size containing the vertices of $W$. The minimum cardinality of a Steiner set of $G$ is the Steiner number of $G$ and it is denoted by $s(G)$. Let $G$ and $H$ be two graphs and let $n$ be the order o","authors_text":"Ismael G. Yero, Juan A. Rodriguez-Velazquez","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-11-15T12:02:28Z","title":"Analogies between the geodetic number and the Steiner number of some classes of graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.3512","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":"sha256:7a484ad04d68e94c3b08ddff7393946c077ede75762cbe50377062a31dbccef4","target":"record","created_at":"2026-05-18T03:58:39Z","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":"09ca263c4fc9994e11d9ac427d4bcd47078e364341f80b1cff94e8c945b45f70","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-11-15T12:02:28Z","title_canon_sha256":"6b4ba5f087a36609c7b2c370f27143f213daf56cb91fe124293cd6eeae69bb7b"},"schema_version":"1.0","source":{"id":"1111.3512","kind":"arxiv","version":2}},"canonical_sha256":"5945eed8f249d571d0460ee98248323440fa9d6d22c41702ed5a2beafe7c0a31","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5945eed8f249d571d0460ee98248323440fa9d6d22c41702ed5a2beafe7c0a31","first_computed_at":"2026-05-18T03:58:39.477632Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:58:39.477632Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZHv4EjljRDcl4xM4YUrdxWJKKddiMqg+l6d3d73w3KQXIZ6mE2p6X59ze08G/1Ps3Sx+Bb2MxSslpyraGkQHCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:58:39.478120Z","signed_message":"canonical_sha256_bytes"},"source_id":"1111.3512","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7a484ad04d68e94c3b08ddff7393946c077ede75762cbe50377062a31dbccef4","sha256:f3e0ab72c5aeb77454f98ad0d25d2afabe7964b95eeb98b4584f92e4e52177f8"],"state_sha256":"8d6d20be8355333e7d5d526d931862d548e9e9f3bb6a6b73af52f16c88595eca"}