{"paper":{"title":"Dynamic Monopolies for Degree Proportional Thresholds in Connected Graphs of Girth at least Five and Trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dieter Rautenbach, Michael Gentner","submitted_at":"2016-01-09T12:06:15Z","abstract_excerpt":"Let $G$ be a graph, and let $\\rho\\in (0,1)$. For a set $D$ of vertices of $G$, let the set $H_{\\rho}(D)$ arise by starting with the set $D$, and iteratively adding further vertices $u$ to the current set if they have at least $\\lceil \\rho d_G(u)\\rceil$ neighbors in it. If $H_{\\rho}(D)$ contains all vertices of $G$, then $D$ is known as an irreversible dynamic monopoly or a perfect target set associated with the threshold function $u\\mapsto \\lceil \\rho d_G(u)\\rceil$. Let $h_{\\rho}(G)$ be the minimum cardinality of such an irreversible dynamic monopoly.\n  For a connected graph $G$ of maximum deg"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.02099","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"}