Let *G = (V,E)* be a graph and *t,r* be positive integers. The *signal* that a tower vertex *T* of signal strength *t* supplies to a vertex *v* is defined as *sig(T, v) = max(t − dist(T,v),0)*, where *dist(T,v)* denotes the distance between the vertices *v* and *T*. In 2015 Blessing, Insko, Johnson, and Mauretour defined a *(t, r) broadcast dominating set*, or simply a *(t, r) broadcast*, on *G* as a set *T ⊆ V* such that the sum of all signal received at each vertex *v ∈ V* from the set of towers *T* is at least *r.* The *(t, r)* broadcast domination number of a finite graph *G*, denoted *γ*_{t,r}(G), is the minimum cardinality over all *(t,r)* broadcasts for G.
Recent research has focused on bounding the *(t, r)* broadcast domination number for the *m×n* grid graph *G*_{m,n}. In 2014, Grez and Farina bounded the k-distance domination number for grid graphs, equivalent to bounding *γ*_{t,1}(G_{m,n}). In 2015, Blessing et al. established bounds on *γ*_{2,2}(G_{m,n}), *γ*_{3,2}(G_{m,n}), and *γ*_{3,3}(G_{m,n}). In this paper, we take the next step and provide a tight upper bound on *γ*_{t,2}(G_{m,n}) for all *t > 2*. We also prove the conjecture of Blessing et al. that their bound on *γ*_{3,2}(G_{m,n}) is tight for large values of *m* and *n*.

]]>