{"paper":{"title":"Interplay between the local metric dimension and 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":"2024-12-22T15:54:50Z","abstract_excerpt":"The local metric dimension ${\\rm dim}_l$ in relation to the clique number $\\omega$ is investigated. It is proved that if $\\omega(G)\\leq n(G)-3$, then ${\\rm dim}_l(G) \\leq n(G)-3$ and the graphs attaining the bound classified. Moreover, the graphs $G$ with ${\\rm dim}_l(G) = n(G)-3$ are listed (with no condition on the clique number). It is proved that if $\\omega(G)=n(G)-2$, then $n(G)-4 \\leq {\\rm dim}_l(G)\\leq n(G)-3$, and all graphs are divided into two groups depending on which of the options applies. The conjecture asserting that for any graph $G$ we have ${\\rm dim}_l(G) \\leq \\left[(\\omega(G"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.17074","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/2412.17074/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"}