{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:QEBBV7XHOVZGUVG2PV3FIZLEL5","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":"1567e6023c10b4ebba6e0bf5ea4a07d682a987b3dceacf9804a200fcb430173a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-11-24T22:29:40Z","title_canon_sha256":"419208407b0d2d4b0077147ccf22c0eac0c8837557ae91bb2c5c2906d148238c"},"schema_version":"1.0","source":{"id":"1111.5864","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.5864","created_at":"2026-05-18T03:03:55Z"},{"alias_kind":"arxiv_version","alias_value":"1111.5864v3","created_at":"2026-05-18T03:03:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.5864","created_at":"2026-05-18T03:03:55Z"},{"alias_kind":"pith_short_12","alias_value":"QEBBV7XHOVZG","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_16","alias_value":"QEBBV7XHOVZGUVG2","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_8","alias_value":"QEBBV7XH","created_at":"2026-05-18T12:26:39Z"}],"graph_snapshots":[{"event_id":"sha256:e4a90b12ff8873b85dd2db22a7cbd2baef46efe93930f29c533347f78b1cdb90","target":"graph","created_at":"2026-05-18T03:03:55Z","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 \\emph{metric dimension} of a graph $G$, denoted by $\\dim(G)$, is the minimum number of vertices such that each vertex is uniquely determined by its distances to the chosen vertices. Let $G_1$ and $G_2$ be disjoint copies of a graph $G$ and let $f: V(G_1) \\rightarrow V(G_2)$ be a function. Then a \\emph{functigraph} $C(G, f)=(V, E)$ has the vertex set $V=V(G_1) \\cup V(G_2)$ and the edge set $E=E(G_1) \\cup E(G_2) \\cup \\{uv \\mid v=f(u)\\}$. We study how metric dimension behaves in passing from $G$ to $C(G,f)$ by first showing that $2 \\le \\dim(C(G, f)) \\le 2n-3$, if $G$ is a connected graph of o","authors_text":"Cong X. Kang, Eunjeong Yi, Linda Eroh","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-11-24T22:29:40Z","title":"On Metric Dimension of Functigraphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.5864","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:f213f63399c273a7c6a6cf92ad273f54c928884c05a646f24f1f675b2e947367","target":"record","created_at":"2026-05-18T03:03:55Z","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":"1567e6023c10b4ebba6e0bf5ea4a07d682a987b3dceacf9804a200fcb430173a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-11-24T22:29:40Z","title_canon_sha256":"419208407b0d2d4b0077147ccf22c0eac0c8837557ae91bb2c5c2906d148238c"},"schema_version":"1.0","source":{"id":"1111.5864","kind":"arxiv","version":3}},"canonical_sha256":"81021afee775726a54da7d765465645f76ba7eb492d46a733f2270075a0da752","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"81021afee775726a54da7d765465645f76ba7eb492d46a733f2270075a0da752","first_computed_at":"2026-05-18T03:03:55.344599Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:03:55.344599Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UcJNMxhqNq7GwqHSEN0IzqJ4/RhXaCx/fS9hVGZ1c55AjMCLNBsEGRXbCRbQvQqpdUirhTSh3bGF0otmH6IGDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:03:55.345242Z","signed_message":"canonical_sha256_bytes"},"source_id":"1111.5864","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f213f63399c273a7c6a6cf92ad273f54c928884c05a646f24f1f675b2e947367","sha256:e4a90b12ff8873b85dd2db22a7cbd2baef46efe93930f29c533347f78b1cdb90"],"state_sha256":"c5eb34efe8d2c0cd71734e3f36f0494adf85c44e9648c584f6ad2f6bfdd6c354"}