A sufficient condition for graphs to be [formula omitted]-optimal

Abstract

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,X̄]|:|X|=k,G[X] is connected}. G is λk-optimal if λk(G)=ξk(G). In 2004, Hellwig and Volkmann gave a sufficient condition for λ2-optimality in graphs of diameter 2. In this paper, we extend the result and give a similar sufficient condition for λk-optimality in graphs of diameter 2 with k≥3.