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Among trees, exactly paths and stars attain the bound, and among connected unicyclic graphs such graphs are $t$-cycles for $t\\in \\{3,4,5\\}$. It is shown that for any $1\\leq n< m$, there exists a graph $G$ with $D(G)=n$ and ${\\rm dim}(G)=m$. Using the bound $D(G) \\le {\\rm dim}(G)+1$, graphs with $D(G) = n(G)-2$ are classified."},"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":"2412.15781","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-12-20T10:48:48Z","cross_cats_sorted":[],"title_canon_sha256":"cf8191ccd0337fb1c0b4747a943b73cd256f14835d1a0e5ac382c37b7bc72402","abstract_canon_sha256":"d8c9adf6c3ac9aa33e119ad51b47dfdcae42535b4efbdcd428a48a92a5797997"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:32:22.054366Z","signature_b64":"/9DIZNHeieyEQHUI2To2DFs4wlbTJ6TPTBnEPAmeTqFe9h9syzI2UhI428YWpQRRDZnFnzpzfLWDAVWwO0moBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9ab2d0f26b39073d81093a350d10f650bd2f0c7e9c7a7e99819c0d5b5221cb46","last_reissued_at":"2026-07-05T11:32:22.053873Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:32:22.053873Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Breaking Symmetry in Graphs by Resolving Sets","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Meysam Korivand, Nasrin Soltankhah, Sandi Klav\\v{z}ar","submitted_at":"2024-12-20T10:48:48Z","abstract_excerpt":"Let ${\\rm dim}(G)$ and $D(G)$ respectively denote the metric dimension and the distinguishing number of a graph $G$. 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