Minimally restricted edge connected graphs

Abstract

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