{"paper":{"title":"Intertwining local (adjacency) metric dimension with the clique number of a graph","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ali Ghalavand, Sandi Klav\\v{z}ar, Xueliang Li","submitted_at":"2025-07-18T09:36:29Z","abstract_excerpt":"Let $G$ be a simple connected graph with order $ n(G)$, local metric dimension $ {\\rm dim}_l(G)$, local adjacency metric dimension $ {\\rm dim}_{A,l}(G)$, and clique number $ \\omega(G)$, where $G\\not\\cong K_{n(G)}$ and $\\omega(G)\\geq3$. It is proved that $ {\\rm dim}_{A,l}(G) \\leq \\left\\lfloor \\left(\\frac{\\omega(G) - 2}{\\omega(G) - 1}\\right)n(G)\\right\\rfloor$. Consequently, the conjecture asserting that the latter expression is an upper bound for ${\\rm dim}_l(G)$ is confirmed. It is important to note that there are infinitely many graphs that satisfy the equalities."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.13777","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.13777/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}