{"paper":{"title":"On the largest dynamic monopolies of graphs with a given average threshold","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kaveh Khoshkhah, Manouchehr Zaker","submitted_at":"2014-05-23T17:38:31Z","abstract_excerpt":"Let $G$ be a graph and $\\tau$ be an assignment of nonnegative integer thresholds to the vertices of $G$. A subset of vertices $D$ is said to be a $\\tau$-dynamic monopoly, if $V(G)$ can be partitioned into subsets $D_0, D_1, \\ldots, D_k$ such that $D_0=D$ and for any $i\\in \\{0, \\ldots, k-1\\}$, each vertex $v$ in $D_{i+1}$ has at least $\\tau(v)$ neighbors in $D_0\\cup \\ldots \\cup D_i$. Denote the size of smallest $\\tau$-dynamic monopoly by $dyn_{\\tau}(G)$ and the average of thresholds in $\\tau$ by $\\overline{\\tau}$. We show that the values of $dyn_{\\tau}(G)$ over all assignments $\\tau$ with the s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.6138","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"}