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Let $G=( V(G), E(G) )$ be a connected graph with vertex set $V(G)$ and edgeset $E(G)$. We say a subset $D$ of $V(G)$ dominates $G$ if every vertex in $V\setminus D$ is adjacent to a vertex in $D$. A generalization of this conceptis $(t,r)$ broadcast domination. We designate certain vertices to be towers ofsignal strength $t$, which send out signal to neighboring vertices with signalstrength decaying linearly as the signal traverses the edges of the graph. Welet $\mathbb{T}$ be the set of all towers, and we define the signal received bya vertex $v\in V(G)$ from a tower $w \in \mathbb T$ to be $f(v)=\sum_{w\in\mathbb{T}}max(0,t-d(v,w))$. Blessing, Insko, Johnson, Mauretour (2014) defineda $(t,r)$ broadcast dominating set, or a $(t,r) $ broadcast, on $G$ as a set$\mathbb{T} \subseteq V(G) $ such that $f(v)\geq r$ for all $v\in V(G)$. Theminimal cardinality of a $(t, r)$ broadcast on $G$ is called the $(t, r)$broadcast domination number of $G$. In this paper, we present our research onthe $(t,r)$ broadcast domination number for certain graphs including paths,grid graphs, the slant lattice, and the king's lattice.

On $(t,r)$ broadcast domination of certain grid graphs