Sufficient conditions for graphs to be λ′-optimal and super-λ′

An edge-cut S of a connected graph G is called a restricted edge-cut if G-S contains no isolated vertices. The minimum cardinality of all restricted edge-cuts is called the restricted edge-connectivity lambda(G) of G. A graph G is said to be lambda'-optimal if lambda'(G) = xi(G), where (G) is the minimum edge-degree of G. A graph is said to be super-lambda' if every minimum restricted edge-cut isolates an edge. In this paper, first, we improve and generalize the sufficient conditions for lambda'-optimality in arbitrary graphs, bipartite graphs, and graphs with diameter 2, which were given by Hellwig and Volkmann, and show using examples that our results are best possible. Second, we provide a simple proof with less restrictive conditions than in Hellwig and Volkmann's theorem that gives sufficient conditions for lambda'-optimality in bipartite graphs. We conclude by presenting sufficient conditions for arbitrary, bipartite, and triangle-free graphs, and for graphs with diameter 2, to be super-lambda' respectively, and demonstrate that these conditions are best possible. (c) 2007 Wiley Periodicals, Inc.