Minimally 3-restricted edge connected graphs

Abstract

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 λ 3 ( G ) . A graph G is called minimally 3-restricted edge connected if λ 3 ( G − e ) < λ 3 ( G ) for each edge e ∈ E . A graph G is λ 3 -optimal if λ 3 ( G ) = ξ 3 ( G ) , where ξ 3 ( G ) = max { ω ( U ) : U ⊂ V ( G ) , G [ U ] is connected , | U | = 3 } , 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 λ 3 -optimal except the 3-cube.