Minimally 3-restricted edge connected graphs

For a connected graph G=(V,E), an edge set S⊂E is a 3-restricted edge cut if G−S is disconnected and every component of G−S has order at least three. The cardinality of a minimum 3-restricted edge cut of G is the 3-restricted edge connectivity of G, denoted by λ₃(G). A graph G is called minimally 3-restricted edge connected if λ₃(G−e)<λ₃(G) for each edge e∈E. A graph G is λ₃-optimal if λ₃(G)=ξ₃(G), where , ω(U) is the number of edges between U and V?U, and G[U] is the subgraph of G induced by vertex set U. We show in this paper that a minimally 3-restricted edge connected graph is always λ₃-optimal except the 3-cube.