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 w S ( v ) = ∑ u ∈ S 1 2 d ¯ ( u , v ) − 1 if v ⁄ ∈ S , and w S ( v ) = 2 if v ∈ S . If, for each v ∈ V ( G ) , we have w S ( v ) ≥ 1 , then S is an exponential dominating set. The smallest cardinality of an exponential dominating set is the exponential domination number, γ e ( G ) . In this paper, we prove: (i) that if G is a connected graph of diameter d , then γ e ( G ) ≥ ( d + 2 ) / 4 , and, (ii) that if G is a connected graph of order n , then γ e ( G ) ≤ 2 5 ( n + 2 ) .