Sufficient conditions for k-restricted edge connected graphs

For a connected graph G=(V,E)G=(V,E), an edge set S⊆ES⊆E is a k  -restricted edge cut if G−SG−S is disconnected and every component of G−SG−S has at least k vertices. The k-restricted edge connectivity of G  , denoted by λk(G)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 (λkλk-optimal for short) if λk(G)=ξk(G)λk(G)=ξk(G). The k  -restricted edge connectivity is more refined network reliability indices than edge connectivity. In this paper, let k≥2k≥2 be an integer, and let G   be a graph of order ν(G)ν(G) at least 2k   satisfying |N(u)∩N(v)|≥2k−2|N(u)∩N(v)|≥2k−2 for all pairs u,vu,v of nonadjacent vertices. If for each triangle T   there exists at least one vertex v∈V(T)v∈V(T) such that d(v)≥⌊ν(G)2⌋+k−1, then G   is λkλk-optimal.