Minimally 3-restricted edge connected graphs

For a connected graph G=(V,E)G=(V,E), an edge set S⊂ES⊂E is a 3-restricted edge cut if G−SG−S is disconnected and every component of G−SG−S has order at least three. The cardinality of a minimum 3-restricted edge cut of GG is the 3-restricted edge connectivity of GG, denoted by λ3(G)λ3(G). A graph GG is called minimally 3-restricted edge connected if λ3(G−e)<λ3(G)λ3(G−e)<λ3(G) for each edge e∈Ee∈E. A graph GG is λ3λ3-optimal if λ3(G)=ξ3(G)λ3(G)=ξ3(G), where ξ3(G)=max{ω(U):U⊂V(G),G[U] is connected,|U|=3}, ω(U)ω(U) is the number of edges between UU and V∖UV∖U, and G[U]G[U] is the subgraph of GG induced by vertex set UU. We show in this paper that a minimally 3-restricted edge connected graph is always λ3λ3-optimal except the 3-cube.