A SUFFICIENT CONDITION FOR GRAPHS TO BE SUPER k-RESTRICTED EDGE CONNECTED

For a subset S of edges in a connected graph G, S 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 lambda(k)(G), is defined as the cardinality of a minimum k-restricted edge cut. Let xi(k)(G) = min{|[X, X]| : |X| = k, G[X] is connected}, where X = V (G)\X. A graph G is super k-restricted edge connected if every minimum k-restricted edge cut of G isolates a component of order exactly k. Let k be a positive integer and let G be a graph of order v >= 2k. In this paper, we show that if |N(u) boolean AND N(v)| >= k +1 for all pairs u, v of nonadjacent vertices and xi(k)(G) <= [v/2] | k , then G is super k-restricted edge connected.