A neighborhood condition for graphs to be maximally k-restricted edge connected

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 λ k ( G ) , is defined as the cardinality of a minimum k-restricted edge cut. Let ξ k ( G ) = min { | [ X , X ¯ ] | : | X | = k , G [ X ] is connected } . G is λ k -optimal if λ k ( G ) = ξ k ( G ) . Let k ⩾ 4 be an integer. In this paper, we show that if | N G ( u ) ∩ N G ( v ) | ⩾ k for all pairs u , v of non-adjacent vertices and ξ k ( G ) ⩽ ⌊ | V | 2 ⌋ + k , then G is λ k -optimal.