{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:QEJNAN2ORES5HGBCA7NOHEXKAP","short_pith_number":"pith:QEJNAN2O","schema_version":"1.0","canonical_sha256":"8112d0374e8925d3982207dae392ea03d4d9cf6dd7a355f3cd791401fd2a33fe","source":{"kind":"arxiv","id":"1103.3336","version":1},"attestation_state":"computed","paper":{"title":"The Metric Dimension of Lexicographic Product of Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Behnaz Omoomi, Mohsen Jannesari","submitted_at":"2011-03-17T05:29:26Z","abstract_excerpt":"For an ordered set $W=\\{w_1,w_2,...,w_k\\}$ of vertices and a vertex $v$ in a connected graph $G$, the ordered $k$-vector $r(v|W):=(d(v,w_1),d(v,w_2),...,d(v,w_k))$ is called the (metric) representation of $v$ with respect to $W$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. The set $W$ is called a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. The minimum cardinality of a resolving set for $G$ is its metric dimension. In this paper, we study the metric dimension of the lexicographic product of graphs $G$ and $H$, $G[H]$."},"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":"1103.3336","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-03-17T05:29:26Z","cross_cats_sorted":[],"title_canon_sha256":"5ba690a1d7a893a20100da0e7b667674c0a48aef5c6d3471246c90aa034a9c51","abstract_canon_sha256":"5c5043ae61adf4b8330f8e70e2caf0e072f8ae9679e0e0a98936bc28508b981c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:26:33.937032Z","signature_b64":"hojjbhRqcrPu8Acu0HfX00kKqO3Xs12Il2A0vHOPRdHGOBOJeLVOC8D89lzRsdSa+wV5z+3tsA/r12mAh/71BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8112d0374e8925d3982207dae392ea03d4d9cf6dd7a355f3cd791401fd2a33fe","last_reissued_at":"2026-05-18T04:26:33.936442Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:26:33.936442Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Metric Dimension of Lexicographic Product of Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Behnaz Omoomi, Mohsen Jannesari","submitted_at":"2011-03-17T05:29:26Z","abstract_excerpt":"For an ordered set $W=\\{w_1,w_2,...,w_k\\}$ of vertices and a vertex $v$ in a connected graph $G$, the ordered $k$-vector $r(v|W):=(d(v,w_1),d(v,w_2),...,d(v,w_k))$ is called the (metric) representation of $v$ with respect to $W$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. The set $W$ is called a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. The minimum cardinality of a resolving set for $G$ is its metric dimension. In this paper, we study the metric dimension of the lexicographic product of graphs $G$ and $H$, $G[H]$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.3336","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":"1103.3336","created_at":"2026-05-18T04:26:33.936550+00:00"},{"alias_kind":"arxiv_version","alias_value":"1103.3336v1","created_at":"2026-05-18T04:26:33.936550+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.3336","created_at":"2026-05-18T04:26:33.936550+00:00"},{"alias_kind":"pith_short_12","alias_value":"QEJNAN2ORES5","created_at":"2026-05-18T12:26:39.201973+00:00"},{"alias_kind":"pith_short_16","alias_value":"QEJNAN2ORES5HGBC","created_at":"2026-05-18T12:26:39.201973+00:00"},{"alias_kind":"pith_short_8","alias_value":"QEJNAN2O","created_at":"2026-05-18T12:26:39.201973+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/QEJNAN2ORES5HGBCA7NOHEXKAP","json":"https://pith.science/pith/QEJNAN2ORES5HGBCA7NOHEXKAP.json","graph_json":"https://pith.science/api/pith-number/QEJNAN2ORES5HGBCA7NOHEXKAP/graph.json","events_json":"https://pith.science/api/pith-number/QEJNAN2ORES5HGBCA7NOHEXKAP/events.json","paper":"https://pith.science/paper/QEJNAN2O"},"agent_actions":{"view_html":"https://pith.science/pith/QEJNAN2ORES5HGBCA7NOHEXKAP","download_json":"https://pith.science/pith/QEJNAN2ORES5HGBCA7NOHEXKAP.json","view_paper":"https://pith.science/paper/QEJNAN2O","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1103.3336&json=true","fetch_graph":"https://pith.science/api/pith-number/QEJNAN2ORES5HGBCA7NOHEXKAP/graph.json","fetch_events":"https://pith.science/api/pith-number/QEJNAN2ORES5HGBCA7NOHEXKAP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QEJNAN2ORES5HGBCA7NOHEXKAP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QEJNAN2ORES5HGBCA7NOHEXKAP/action/storage_attestation","attest_author":"https://pith.science/pith/QEJNAN2ORES5HGBCA7NOHEXKAP/action/author_attestation","sign_citation":"https://pith.science/pith/QEJNAN2ORES5HGBCA7NOHEXKAP/action/citation_signature","submit_replication":"https://pith.science/pith/QEJNAN2ORES5HGBCA7NOHEXKAP/action/replication_record"}},"created_at":"2026-05-18T04:26:33.936550+00:00","updated_at":"2026-05-18T04:26:33.936550+00:00"}