Home > RHUMJ > Vol. 20 (2019) > Iss. 1

#### Abstract

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*. In 2015, Blessing et al. established bounds on

_{t,1}(G_{m,n})*γ*,

_{2,2}(G_{m,n})*γ*, and

_{3,2}(G_{m,n})*γ*. In this paper, we take the next step and provide a tight upper bound on

_{3,3}(G_{m,n})*γ*for all

_{t,2}(G_{m,n})*t > 2*. We also prove the conjecture of Blessing et al. that their bound on

*γ*is tight for large values of

_{3,2}(G_{m,n})*m*and

*n*.

#### Sponsor

Pamela E. Harris (peh2@williams.edu)

#### Recommended Citation

Randolph, Timothy W.
(2019)
"Asymptotically Optimal Bounds for (t,2) Broadcast Domination on Finite Grids,"
*Rose-Hulman Undergraduate Mathematics Journal*: Vol. 20
:
Iss.
1
, Article 5.

Available at:
https://scholar.rose-hulman.edu/rhumj/vol20/iss1/5