broadcast domination in tori

A broadcast on a graph G is a function f : V (G) → {0, 1, . . . , diam(G)} such that for every vertex v ∈ V (G), f (v) ≤ e(v), where diam(G) is the diameter of G, and e(v) is the eccentricity of v. In addition, if every vertex hears the broadcast, then the broadcast is a dominating broadcast. The cost of a broadcast f is the value σ(f ) = ∑v∈V (G) f (v). In this paper we determine the minimum cost of a dominating broadcast (also known as the broadcast domination number) for a torus Cm □ Cn.