The vertex connectivity of a graph G is denoted by κ(G) and the minimum degree of G is denoted by δ(G). A finite simple graph G is said to be critically (k, k)-connected if and for each vertex ν of Gκ(G − ν) = k − 1 or , where is the complement of G. The following result is proved: If G is acritically (k, k)-connected graphs, and , then |V(G)|⩽4k. Furthermore, these bounds are sharp for k ⩾ 3.