{"paper":{"title":"Some Comments on the Slater number","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dieter Rautenbach, Michael Gentner","submitted_at":"2016-08-16T11:54:29Z","abstract_excerpt":"Let $G$ be a graph with degree sequence $d_1\\geq \\ldots \\geq d_n$. Slater proposed $s\\ell(G)=\\min\\{ s: (d_1+1)+\\cdots+(d_s+1)\\geq n\\}$ as a lower bound on the domination number $\\gamma(G)$ of $G$. We show that deciding the equality of $\\gamma(G)$ and $s\\ell(G)$ for a given graph $G$ is NP-complete but that one can decide efficiently whether $\\gamma(G)>s\\ell(G)$ or $\\gamma(G)\\leq \\left(\\left\\lceil\\ln \\left(\\frac{n(G)}{s\\ell(G)}\\right)\\right\\rceil+1\\right)s\\ell(G)$. For real numbers $\\alpha$ and $\\beta$ with $\\alpha\\geq \\max\\{ 0,\\beta\\}$, let ${\\cal G}(\\alpha,\\beta)$ be the class of non-null gra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.04560","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"}