On locally k-critically n-connected graphs

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Let ∅ ≠ W ⊆ V( G). The graph G is called a W-locally k-critically n-connected graph or simply a W-locally ( n, k)-graph, if for all V′ ⊆ W with | V′|⩽ k and each fragment F of G we have that κ( G− V′)= n−| V′| and F ∩ W ≠ ∅. In this paper we prove that every non-complete W-locally ( n, k)-graph has (2 k+2) distinct fragments and | W|⩾2 k+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 (2 k+2) pairwise disjoint ends. (2) Slater's conjecture on ( n, k)-graphs holds, i.e., the complete graph K n+1 is the unique ( n, k)-graph for 2 k> n.