Domination with exponential decay

Let G be a graph and S ⊆ V (G). For each vertex u ∈ S and for each v ∈ V (G) − S, we define d(u, v) = d(v, u) to be the length of a shortest path in 〈V (G)−(S−{u})〉 if such a path exists, and∞ otherwise. Let v ∈ V (G). We define wS(v) = ∑ u∈S 1 2d(u,v)−1 if v 6∈ S, and wS(v) = 2 if v ∈ S. If, for each v ∈ V (G), we have wS(v) ≥ 1, then S is an exponential dominating set. The smallest cardinality of an exponential dominating set is the exponential domination number, γe(G). In this paper, we prove: (i) that if G is a connected graph of diameter d, then γe(G) ≥ (d + 2)/4, and, (ii) that if G is a connected graph of order n, then γe(G) ≤ 25(n + 2).