Sufficient conditions for graphs to be maximally 4-restricted edge connected

For a subset S of edges in a connected graph G, the set S 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. A connected graph G is said to be λ_k-connected if G has a k-restricted edge cut. Let ξ_k(G) = min{|[X, ̅X ]| : |X| = k, G[X] is connected}, where ̅X = V (G)X. A graph G is said to be maximally k-restricted edge connected if λ_k(G) = ξ_k(G). In this paper we show that if G is a λ₄-connected graph with λ₄(G) ≤ ξ₄(G) and the girth satisfies g(G) ≥ 8, and there do not exist six vertices u₁, u₂, u₃, v₁, v₂ and v₃ in G such that the distance d(u_i, v_j) ≥ 3, (1 ≤ i, j ≤ 3), then G is maximally 4-restricted edge connected.