{"paper":{"title":"Upper broadcast domination of toroidal grids and a classification of diametrical trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bethany Kubik, Candice Price, Erik Insko","submitted_at":"2016-10-17T18:27:29Z","abstract_excerpt":"A broadcast on a graph $G=(V,E)$ is a function $f:V \\rightarrow \\{0,1, \\ldots, \\text{diam}(G)\\}$ satisfying $f(v) \\leq e(v)$ for all $v \\in V$, where $e(v)$ denotes the eccentricity of $v$ and $\\text{diam}(G)$ denotes the diameter of $G$. We say that a broadcast dominates $G$ if every vertex can hear at least one broadcasting node. The upper domination number is the maximum cost of all possible minimal broadcasts, where the cost of a broadcast is defined as $\\text{cost} (f)= \\sum_{v \\in V}f(v)$. In this paper we establish both the upper domination number and the upper broadcast domination numb"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.05250","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"}