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Bounds On$(t,r)$Broadcast Domination of$n$ -Dimensional Grids
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Bounds On$(t,r)$Broadcast Domination of$n$ -Dimensional Grids
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Journal Article
Bounds On$(t,r)$Broadcast Domination of$n$ -Dimensional Grids
2023
Overview
In this paper, we study a variant of graph domination known as$(t, r)$broadcast domination, first defined in Blessing, Insko, Johnson, and Mauretour in 2015. In this variant, each broadcast provides$t-d$reception to each vertex a distance$d < t$from the broadcast. If$d \\ge t$then no reception is provided. A vertex is considered dominated if it receives$r$total reception from all broadcasts. Our main results provide some upper and lower bounds on the density of a$(t, r)$dominating pattern of an infinite grid, as well as methods of computing them. Also, when$r \\ge 2$we describe a family of counterexamples to a generalization of Vizing's Conjecture to$(t,r)$broadcast domination.
Publisher
Discrete Mathematics & Theoretical Computer Science
