Optimal (t, r) broadcasts on the infinite grid

Let G = (V, E) be a graph and t, r be positive integers. The reception strength (or signal) that a vertex upsilon receives from a tower of signal strength t located at vertex T is defined as sig(upsilon, T) = max(t - dist(upsilon, T), 0), where dist(upsilon, T) denotes the distance between the vertices upsilon 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 subset of V such that the sum of all signals received at each vertex upsilon is an element of V is at least r. We say that T is optimal if vertical bar T vertical bar 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) is an element of {(2, 2), (3, 1), (3, 2), (3, 3)1 and gave optimal (t, r) broadcasts on G(m,n), the grid graph of dimension m x 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. (C) 2018 Elsevier B.V. All rights reserved.