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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"},"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":"1111.5864","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-11-24T22:29:40Z","cross_cats_sorted":[],"title_canon_sha256":"419208407b0d2d4b0077147ccf22c0eac0c8837557ae91bb2c5c2906d148238c","abstract_canon_sha256":"1567e6023c10b4ebba6e0bf5ea4a07d682a987b3dceacf9804a200fcb430173a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:03:55.345242Z","signature_b64":"UcJNMxhqNq7GwqHSEN0IzqJ4/RhXaCx/fS9hVGZ1c55AjMCLNBsEGRXbCRbQvQqpdUirhTSh3bGF0otmH6IGDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"81021afee775726a54da7d765465645f76ba7eb492d46a733f2270075a0da752","last_reissued_at":"2026-05-18T03:03:55.344599Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:03:55.344599Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Metric Dimension of Functigraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Cong X. Kang, Eunjeong Yi, Linda Eroh","submitted_at":"2011-11-24T22:29:40Z","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 $E=E(G_1) \\cup E(G_2) \\cup \\{uv \\mid v=f(u)\\}$. 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