ON LOCALLY K-CRITICALLY N-CONNECTED GRAPHS

Let 0 not-equal W subset-or-equal-to V(G). The graph G is called a W-l ocally k-critically n-connected graph or simply a W-locally (n, k)-gra ph, if for all V' subset-or-equal-to W with \V'\ less-than-or-equal-to k and each fragment F of G we have that K(G - V') = n - \V'\ and F an d W not-equal 0. In this paper we prove that every non-complete W-loca lly (n, k)-graph has (2k + 2) distinct fragments and Absolute value of W greater-than-or-equal-to 2k + 2. From this result it follows that: (1) Let G be a non-complete (n, k)-graph. If all ends of G are proper, then G has (2k + 2) pairwise disjoint ends. (2) Slater's conjecture o n (n, k)-graphs holds, i.e., the complete graph K(n + 1) is the unique (n, k)-graph for 2k > n.