A fragment of a connected graph G is a subset A of V(G) consisting of components of G - S such that V(G) - S - A ≠ 0 where S is a minimum cut of G. A graph G is said to be (k, k)-connected if its connectivity κ(G) = k and κ(G) = k. A fragment A of a (k,k )-connected graphG is called elemental if A < G - (k + k). A (k, k)-connected graph G is said to be critical if κ(G - x) = k - 1 or κ(G - x) = k - 1 for each vertex x in V(G). We prove the following result.Let k and k be integers with 1 k k, and let G be a critically (k, k)-connected graph. If no minimum cut of G (resp. G) contains all the elemental fragments of G (resp. G), then G (k + 1) (k + 1) + kk2 - 1 k2.Moreover, for any given positive integers k and k, there is a critically (k, k)-connected graph for which the above equality holds.