Optimal [formula omitted] broadcasts on the infinite grid

Abstract

Let G=(V,E) be a graph and t,r be positive integers. The reception strength (or signal) that a vertex v receives from a tower of signal strength t located at vertex T is defined as sig(v,T)=max(t−dist(v,T),0), where dist(v,T) 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 signals received at each vertex v∈V is at least r. We say that T is optimal if |T| is minimal among all such sets T. The cardinality of an optimal (t,r) broadcast on a finite graph G is called the (t,r) broadcast domination number of G. The concept of (t,r) broadcast domination generalizes the classical problem of domination on graphs. In fact, the (2,1) broadcasts on a graph G are exactly the dominating sets of G.In their paper, Blessing et al. considered (t,r)∈{(2,2),(3,1),(3,2),(3,3)} and gave optimal (t,r) broadcasts on Gm,n, the grid graph of dimension m×n, for small values of m and n. They also provided upper bounds on the optimal (t,r) broadcast numbers for grid graphs of arbitrary dimensions. In this paper, we define the density of a (t,r) broadcast, which allows us to provide optimal (t,r) broadcasts on the infinite grid graph for all t≥2 and r=1,2, and bound the density of the optimal (t,3) broadcast for all t≥2. In addition, we present a Python program to compute upper bounds on the density of a minimal (t,r) broadcast on the infinite grid, and compute these bounds for all 1≤t≤15 and 1≤r≤40. Lastly, we construct a family of counterexamples to the conjecture of Blessing et al. that the optimal (t,r) and (t+1,r+2) broadcasts are identical for all t≥1 and r≥1 on the infinite grid.