{"paper":{"title":"The strong convexity spectra of grids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"C\\'esar Hern\\'andez-Cruz, Gabriela Araujo-Pardo, Juan Jos\\'e Montellano-Ballesteros","submitted_at":"2017-03-08T00:59:01Z","abstract_excerpt":"Let $D$ be a connected oriented graph. A set $S \\subseteq V(D)$ is convex in $D$ if, for every pair of vertices $x, y \\in S$, the vertex set of every $xy$-geodesic, ($xy$ shortest directed path) and every $yx$-geodesic in $D$ is contained in $S$. The convexity number, ${\\rm con}(D)$, of a non-trivial oriented graph, $D$, is the maximum cardinality of a proper convex set of $D$. The strong convexity spectrum of the graph $G$, $S_{SC} (G)$, is the set $\\{{ \\rm con}(D) \\colon\\ D {\\rm \\ is \\ a \\ strong \\ orientation \\ of \\ } G \\}$. In this paper we prove that the problem of determining the convexi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.02654","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"}