{"paper":{"title":"Extremal Graph Theory for Metric Dimension and Diameter","license":"","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Carlos Seara, Carmen Hernando, David R. Wood, Ignacio M. Pelayo, Merce Mora","submitted_at":"2007-05-07T16:16:12Z","abstract_excerpt":"A set of vertices $S$ \\emph{resolves} a connected graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The \\emph{metric dimension} of $G$ is the minimum cardinality of a resolving set of $G$. Let $\\mathcal{G}_{\\beta,D}$ be the set of graphs with metric dimension $\\beta$ and diameter $D$. It is well-known that the minimum order of a graph in $\\mathcal{G}_{\\beta,D}$ is exactly $\\beta+D$. The first contribution of this paper is to characterise the graphs in $\\mathcal{G}_{\\beta,D}$ with order $\\beta+D$ for all values of $\\beta$ and $D$. Such a charact"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0705.0938","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":""},"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"}