Let be a graph. A vertex strongly dominates a vertex if and . A set is a strong dominating set of if every vertex in is strongly dominated by at least one vertex of .The minimum cardinality among all strong dominating sets of is called the strong domination number of and is denoted by . This parameter was introduced by Sampathkumar and Pushpa Latha in [4].In this paper, we investigate sharp upper bounds on the strong domination number for a tree and a connected graph. We show that for any tree of order that is different from the tree obtained from a star by subdividing each edge once, and this bound is sharp.For any connected graph of order , it is shown that and this bound is sharp. We show that the decision problem corresponding to the computation of is NP-complete, even for bipartite or chordal graphs.