Sufficient conditions for super k-restricted edge connectivity in graphs of diameter 2

For a connected graph G = (V, E), an edge set S ⊆ E is a k-restricted edge cut if G−S is disconnected and every component of G− S has at least k vertices. The k-restricted edge connectivity of G, denoted by λk(G), is defined as the cardinality of a minimum k-restricted edge cut. Let ξk(G) = min{|[X, X ]| : |X | = k,G[X ] is connected}. G is λk -optimal if λk(G) = ξk(G). Moreover, G is super-λk if every minimum k-restricted edge cut of G isolates one connected subgraph of order k. In this paper, we prove that if |NG(u) ∩ NG(v)| ≥ 2k − 1 for all pairs u, v of nonadjacent vertices, then G is λk -optimal; and if |NG(u) ∩ NG(v)| ≥ 2k for all pairs u, v of nonadjacent vertices, then G is either super-λk or in a special class of graphs. In addition, for k-isoperimetric edge connectivity, which is closely related with the concept of k-restricted edge connectivity, we show similar results. c © 2008 Elsevier B.V. All rights reserved.