An upper bound for orders of certain (k, k))-connected graphs

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≠∅ 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 graph G 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) + (k ⌈k2⌉-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.