{"paper":{"title":"On the boundary as an $x$-geodominating set in graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"J. C\\'aceres, M.L. Puertas, M. Morales","submitted_at":"2013-11-15T10:51:06Z","abstract_excerpt":"Given a graph $G$ and a vertex $x\\in V(G)$, a vertex set $S \\subseteq V(G)$ is an $x$-geodominating set of $G$ if each vertex $v\\in V(G)$ lies on an $x-y$ geodesic for some element $y\\in S$. The minimum cardinality of an $x$-geodominating set of $G$ is defined as the $x$-geodomination number of $G$, $g_x(G)$, and an $x$-geodominating set of cardinality $g_x(G)$ is called a $g_x$-set and it is known that it is unique for each vertex $x$. We prove that, in any graph $G$, the $g_x$-set associated to a vertex $x$ is the set of boundary vertices of $x$, that is $\\partial(x)= \\{v \\in V(G) : \\forall "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.3804","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"}