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The minimum $k$ for which there is a resolving $k$-partition of $V$ is the \\emph{partition dimension} of $G$. Salman et al.{\\rm\\cite{SaJaCh12}} claimed that \\em"},"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":"1507.05239","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-19T02:36:43Z","cross_cats_sorted":[],"title_canon_sha256":"e7b384042bb44383adc9235ad566c50cfd02fe832f78b6e9f802a791599bd64b","abstract_canon_sha256":"2fb423d2ef320359130073a040de13b79e1a0281f518639e1920f32c5d405d50"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:01:01.316542Z","signature_b64":"FI7/RIYI30sUXi6HHFvpWUVMldLZTVN8VAoTjGOrAb6TBI0mF7npBLs1CY9l0XQJgPCJpym5m4d1jojwVHKqCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"14984a9355f5e9614d1b3dde821bb8e22018f0e91b48a36102365a889aa37718","last_reissued_at":"2026-05-18T01:01:01.315865Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:01:01.315865Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Partition Dimension of Circulant Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bharati Rajan, Cyriac Grigorious, Mirka Miller, Paul Manuel, Sudeep Stephen","submitted_at":"2015-07-19T02:36:43Z","abstract_excerpt":"For a vertex $v$ of a connected graph $G(V,E)$ and a subset $S$ of $V$, the distance between $v$ and $S$ is defined by $d(v,S)=min\\{d(v,x):x \\in S \\}.$ For an ordered \\emph{k}-partition $\\Pi=\\{S_1,S_2\\ldots S_k\\}$ of $V$, the representation of $v$ with respect to $\\Pi$ is the $k$-vector $r(v|\\Pi) =(d(v,S_1),d(v,S_2)\\ldots d(v,S_k)).$ The $k$-partition $\\Pi$ is a resolving partition if the $k$-vectors $r(v|\\Pi)$, $v \\in V$ are distinct. The minimum $k$ for which there is a resolving $k$-partition of $V$ is the \\emph{partition dimension} of $G$. 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