Geodesic
Bounds On $(t,r)$ Broadcast Domination of $n$-Dimensional Grids
Discrete mathematics & theoretical computer science, Tome 24 (2022) no. 1.

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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.
DOI : 10.46298/dmtcs.5732
Classification : 05C69 
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Shlomi, Tom. Bounds On $(t,r)$ Broadcast Domination of $n$-Dimensional Grids. Discrete mathematics & theoretical computer science, Tome 24 (2022) no. 1. doi : 10.46298/dmtcs.5732. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.5732/

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