Let δ, γ, κ and α be, respectively, the minimum degree, the domination number, the connectivity and the independence number of a graph G. The graph G is 3-domination-critical if γ=3 and the addition of any edge decreases γ by 1. In this paper, we prove that if G is a 3-domination-critical graph, then ακ+2; and moreover, if κδ−1, then ακ+1. We also give a short proof of Wojcicka's result, which says that every connected 3-domination-critical graph of order at least 7 contains a hamiltonian path (J. Graph Theory 14 (1990) 205).