{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:VLNJCBCL3T3ZDNMS4VCKAJDSUZ","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":"efacdfcabdb311dcc8be6e0698369b75991a2fb4e8ced0ecf8db189418a4a7e5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-24T14:54:22Z","title_canon_sha256":"d1fda745903884a4ba677bf123c0068834a63759e4716dda6eb3fd6ec359b1a4"},"schema_version":"1.0","source":{"id":"1602.07537","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.07537","created_at":"2026-05-18T01:12:43Z"},{"alias_kind":"arxiv_version","alias_value":"1602.07537v3","created_at":"2026-05-18T01:12:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.07537","created_at":"2026-05-18T01:12:43Z"},{"alias_kind":"pith_short_12","alias_value":"VLNJCBCL3T3Z","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_16","alias_value":"VLNJCBCL3T3ZDNMS","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_8","alias_value":"VLNJCBCL","created_at":"2026-05-18T12:30:48Z"}],"graph_snapshots":[{"event_id":"sha256:d872c145a9508f818455e6e532a497b8a740485c10b00aef28d118af155b4a07","target":"graph","created_at":"2026-05-18T01:12:43Z","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 metric dimension is quite a well-studied graph parameter. Recently, the adjacency dimension and the local metric dimension have been introduced and studied. In this paper, we give a general formula for the local metric dimension of the lexicographic product $G \\circ \\mathcal{H}$ of a connected graph $G$ of order $n$ and a family $\\mathcal{H}$ composed by $n$ graphs. We show that the local metric dimension of $G \\circ \\mathcal{H}$ can be expressed in terms of the true twin equivalence classes of $G$ and the local adjacency dimension of the graphs in $\\mathcal{H}$.","authors_text":"A. Estrada-Moreno, G. A. Barrag\\'an-Ram\\'irez, J. A. Rodr\\'iguez-Vel\\'azquez, Y. Ram\\'irez-Cruz","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-24T14:54:22Z","title":"The local metric dimension of the lexicographic product of graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.07537","kind":"arxiv","version":3},"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":"sha256:13e24850797380f663dfef2c9b7746ec99be48808bbd0764b447caa026ecc060","target":"record","created_at":"2026-05-18T01:12:43Z","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":"efacdfcabdb311dcc8be6e0698369b75991a2fb4e8ced0ecf8db189418a4a7e5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-24T14:54:22Z","title_canon_sha256":"d1fda745903884a4ba677bf123c0068834a63759e4716dda6eb3fd6ec359b1a4"},"schema_version":"1.0","source":{"id":"1602.07537","kind":"arxiv","version":3}},"canonical_sha256":"aada91044bdcf791b592e544a02472a67464e413920b57f171834308f7add598","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"aada91044bdcf791b592e544a02472a67464e413920b57f171834308f7add598","first_computed_at":"2026-05-18T01:12:43.170168Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:12:43.170168Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Q/WZBDDduL5CakCR+WtEjCvavYgwcXz7WS5u8/ugYeS0U1rNbYiqpngR6q7OEDvUV5AIF9zstjF/LC0EcHBJCw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:12:43.170534Z","signed_message":"canonical_sha256_bytes"},"source_id":"1602.07537","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:13e24850797380f663dfef2c9b7746ec99be48808bbd0764b447caa026ecc060","sha256:d872c145a9508f818455e6e532a497b8a740485c10b00aef28d118af155b4a07"],"state_sha256":"931f693b7a71422a3cf480213e1774963251351cc5056f79e219c26e1e22748a"}