Domination with exponential decay

Let GG be a graph and S⊆V(G)S⊆V(G). For each vertex u∈Su∈S and for each v∈V(G)−Sv∈V(G)−S, we define d¯(u,v)=d¯(v,u) to be the length of a shortest path in 〈V(G)−(S−{u})〉〈V(G)−(S−{u})〉 if such a path exists, and ∞∞ otherwise. Let v∈V(G)v∈V(G). We define wS(v)=∑u∈S12d¯(u,v)−1 if v⁄∈Sv⁄∈S, and wS(v)=2wS(v)=2 if v∈Sv∈S. If, for each v∈V(G)v∈V(G), we have wS(v)≥1wS(v)≥1, then SS is an exponential dominating set. The smallest cardinality of an exponential dominating set is the exponential domination number  , γe(G)γe(G). In this paper, we prove: (i) that if GG is a connected graph of diameter dd, then γe(G)≥(d+2)/4γe(G)≥(d+2)/4, and, (ii) that if GG is a connected graph of order nn, then γe(G)≤25(n+2).