{"paper":{"title":"Optimal $(t,r)$ Broadcasts On the Infinite Grid","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Benjamin F. Drews, Pamela E. Harris, Timothy W. Randolph","submitted_at":"2017-11-29T21:37:05Z","abstract_excerpt":"Let $G=(V,E)$ be a graph and $t,r$ be positive integers. The signal that a vertex $v$ receives from a tower of signal strength $t$ located at vertex $T$ is defined as $sig(v,T)=max(t-dist(v,T),0)$, where $dist(v,T)$ denotes the distance between the vertices $v$ and $T$. In 2015 Blessing, Insko, Johnson, and Mauretour defined a $(t,r)$ broadcast dominating set, or simply a $(t,r)$ broadcast, on $G$ as a set $\\mathbb{T}\\subseteq V$ such that the sum of all signal received at each vertex $v \\in V$ is at least $r$. We say that $\\mathbb{T}$ is optimal if $|\\mathbb{T}|$ is minimal among all such set"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.11116","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"}