Every cubic cage is quasi 4-connected

Throughout this paper, G = (V,E) denotes a (δ, g)-graph with vertex set V and edge set E, that is, a regular simple graph of degree δ and girth g. For every v ∈ V, N(v) denotes the neighbourhood of v. If S ⊂ V, then N(S) = ∪v∈SN(v). If H is a subgraph of G, then NH(S) = N(S) ∩ V (H). The subgraph of G induced by S is denoted G[S]. For X,Y ⊂ V, d(X,Y) = min{d(x, y) : x ∈ X, y ∈ Y}, where d(x, y) denotes the distance between x and y. The diameter D of G is the maximum distance over all pairs of vertices in G. A graph G is called connected if every pair of vertices is joined by a path. If S ⊂ V and G − S is not connected, then G is said to be a cutset. A (connected) component of a nonconnected graph G is a maximal connected subgraph of G. A connected graph is called k-connected if every cutset has cardinality at least k. The connectivity κ of a connected graph G is defined as the maximum integer k such that G is k-connected. The minimum cutsets are those having cardinality κ. A minimum cutset S is called trivial if S = N(v) for some v ∈ V. A (δ, g)-graph G is superconnected if κ = δ and every minimum cutset is trivial (see [4]). A superconnected (3, g)-graph is quasi 4-connected if for each minimum cutset S = N(v), G − S has precisely two components. Quasi 4-connected graphs, which exhibit many of the propertie