Minimally restricted edge connected graphs

For a connected graph G=(V,E)G=(V,E), an edge set S⊂ES⊂E is a restricted edge cut if G−SG−S is disconnected and there is no isolated vertex in G−SG−S. The cardinality of a minimum restricted edge cut of GG is the restricted edge connectivity of GG, denoted by λ′(G)λ′(G). A graph GG is called minimally restricted edge connected if λ′(G−e)<λ′(G)λ′(G−e)<λ′(G) for each edge e∈Ee∈E. A graph GG is λ′λ′-optimal if λ′(G)=ξ(G)λ′(G)=ξ(G), where ξ(G)ξ(G) is the minimum edge degree of GG. We show in this work that a minimally restricted edge connected graph is always λ′λ′-optimal.