Edge-cuts leaving components of order at least three

Let G be a graph with vertex set V(G) and edge set E(G). For X⊆V(G) let G[X] be the subgraph induced by X, X̄=V(G)−X, and (X,X̄) the set of edges in G with one end in X and the other in X̄. If G is a connected graph and S⊂E(G) such that G−S is disconnected, and each component of G−S consists of at least three vertices, then we speak of an order-3 edge-cut. The minimum cardinality |S| over all order-3 edge-cuts in G is called the order-3 edge-connectivity, denoted by λ3=λ3(G). A connected graph G is λ3-connected, if λ3(G) exists. An order-3 edge-cut S in G is called a λ3-cut, if |S|=λ3. First of all, we characterize the class of graphs which are not λ3-connected. Then we show for λ3-connected graphs G that λ3(G)⩽ξ3(G), where ξ3(G) is defined byξ3(G)=min{|(X,X̄)|: X⊂V(G), |X|=3, G[X] is connected}.A λ3-connected graph G is called λ3-optimal, if λ3(G)=ξ3(G). If (X,X̄) is a λ3-cut, then X⊂V(G) is called a λ3-fragment. Letr3(G)=min{|X|: X is a λ3-fragment of G}.We prove that a λ3-connected graph G is λ3-optimal if and only if r3(G)=3. Finally, we study the λ3-optimality of some graph classes. In particular, we show that the complete bipartite graph Kr,s with r,s⩾2 and r+s⩾6 is λ3-optimal.