{"paper":{"title":"On the Decision Number of Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"K. Ehsani, M. Dalirrooyfard, R. Sherkati, S. Akbari, S. Davodpoor","submitted_at":"2014-02-01T22:51:09Z","abstract_excerpt":"Let $G$ be a graph. A good function is a function $f:V(G)\\rightarrow \\{-1,1\\}$, satisfying $f(N(v))\\geq 1$, for each $v\\in V(G)$, where $ N(v)=\\{u\\in V(G)\\, |\\, uv\\in E(G) \\} $ and $f(S) = \\sum_{u\\in S} f(u)$ for every $S \\subseteq V(G) $. For every cubic graph $G$ of order $ n, $ we prove that $ \\gamma(G) \\leq \\frac{5n}{7} $ and show that this inequality is sharp. A function $f:V(G)\\rightarrow \\{-1,1\\}$ is called a nice function, if $f(N[v])\\le1$, for each $v\\in V(G)$, where $ N[v]=\\{v\\} \\cup N(v) $. Define $\\overline{\\beta}(G)=max\\{f(V(G))\\}$, where $f$ is a nice function for $G$. We show th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.0134","kind":"arxiv","version":2},"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"}