  Let $G=( V(G), E(G) )$ be a connected graph with vertex set $V(G)$ and edge
set $E(G)$. 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 concept
is $(t,r)$ broadcast domination. We designate certain vertices to be towers of
signal strength $t$, which send out signal to neighboring vertices with signal
strength decaying linearly as the signal traverses the edges of the graph. We
let $\mathbb{T}$ be the set of all towers, and we define the signal received by
a 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) defined
a $(t,r)$ 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 $v\in V(G)$. The
minimal 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 on
the $(t,r)$ broadcast domination number for certain graphs including paths,
grid graphs, the slant lattice, and the king's lattice.
