Given a graph G = (V, E), set S ⊂ V is a dominating set if each node of V - S is adjacent to at least one node in S. The domination number of G is the smallest size of a dominating set and the codomination number is the domination number of its complement. We determine the codomination number of a graph having diameter at least three. Further we explore the effects of this result on the open problem of characterizing graphs having equal domination and codomination numbers.