{"paper":{"title":"Extremal Problems Related to the Cardinality Redundance of Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Daniel McGinnis, Nathan Shank","submitted_at":"2018-10-19T19:13:33Z","abstract_excerpt":"A dominating set of a graph $G$ is a set of vertices $D$ such that for all $v \\in V(G)$, either $v \\in D$ or $(v,d) \\in E(G)$ for some $d \\in D$. The cardinality redundance of a vertex set $S$, $CR(S)$, is the number of vertices in $V(G)$ such that $|N[x] \\cap S| \\geq 2$. The cardinality redundance of $G$ is the minimum of $CR(S)$ taken over all dominating sets $S$. A set that achieves $CR(G)$ is a $\\gamma_{cr}$-set, and the size of the minimum $\\gamma_{cr}$-set is $\\gamma_{cr}(G)$. We give the maximum number of edges in a graph with a given number of vertices and given cardinality redundance."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.08657","kind":"arxiv","version":4},"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"}