{"paper":{"title":"Extremal Graph Theory for Metric Dimension and Girth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Mohsen Jannesari","submitted_at":"2012-03-07T19:48:21Z","abstract_excerpt":"A set $W\\subseteq V(G)$ is called a resolving set for $G$, if for each two distinct vertices $u,v\\in V(G)$ there exists $w\\in W$ such that $d(u,w)\\neq d(v,w)$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. The minimum cardinality of a resolving set for $G$ is called the metric dimension of $G$, and denoted by $\\beta(G)$. In this paper, it is proved that in a connected graph $G$ of order $n$ which has a cycle, $\\beta(G)\\leq n-g(G)+2$, where $g(G)$ is the length of a shortest cycle in $G$, and the equality holds if and only if $G$ is a cycle, a complete graph or a complete bip"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.1584","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}