Sufficient conditions for k-restricted edge connected graphs

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,X¯]|:|X|=k,G[X]is connected}, where X¯=V\\X. G is maximally k-restricted edge connected (λk-optimal for short) if λk(G)=ξk(G). The k-restricted edge connectivity is more refined network reliability indices than edge connectivity. In this paper, let k≥2 be an integer, and let G be a graph of order ν(G) at least 2k satisfying |N(u)∩N(v)|≥2k−2 for all pairs u,v of nonadjacent vertices. If for each triangle T there exists at least one vertex v∈V(T) such that d(v)≥⌊ν(G)2⌋+k−1, then G is λk-optimal.