{"paper":{"title":"$\\beta$-Packing Sets in Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Benjamin M. Case, Evan M. Haithcock, Renu C. Laskar","submitted_at":"2019-05-31T20:45:25Z","abstract_excerpt":"A set $S\\subseteq V$ is $\\alpha$-dominating if for all $v\\in V-S$, $|N(v) \\cap S | \\geq \\alpha |N(v)|.$ The $\\alpha$-domination number of $G$ equals the minimum cardinality of an $\\alpha$-dominating set $S$ in $G$. Since being introduced by Dunbar, et al. in 2000, $\\alpha$-domination has been studied for various graphs and a variety of bounds have been developed. In this paper, we propose a new parameter derived by flipping the inequality in the definition of $\\alpha$-domination. We say a set $S \\subset V$ is a $\\beta$-packing set of a graph $G$ if $S$ is a proper, maximal set having the prope"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.00073","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"}