Bounds On $(t,r)$ Broadcast Domination of $n$-Dimensional Grids

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