Edge-cuts leaving components of order at least three

Abstract

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, X ̄ =V(G)−X , and (X, X ̄ ) the set of edges in G with one end in X and the other in X ̄ . 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 λ 3= λ 3( G). A connected graph G is λ 3- connected, if λ 3( G) exists. An order-3 edge-cut S in G is called a λ 3- cut, if | S|= λ 3. First of all, we characterize the class of graphs which are not λ 3-connected. Then we show for λ 3-connected graphs G that λ 3( G)⩽ ξ 3( G), where ξ 3( G) is defined by ξ 3(G)= min{|(X, X ̄ )| : X⊂V(G), |X|=3, G[X] is connected }. A λ 3-connected graph G is called λ 3- optimal, if λ 3( G)= ξ 3( G). If (X, X ̄ ) is a λ 3-cut, then X⊂ V( G) is called a λ 3- fragment. Let r 3(G)= min{|X| : X is a λ 3- fragment of G}. We prove that a λ 3-connected graph G is λ 3-optimal if and only if r 3( G)=3. Finally, we study the λ 3-optimality of some graph classes. In particular, we show that the complete bipartite graph K r, s with r, s⩾2 and r+ s⩾6 is λ 3-optimal.