A characterization of weakly four-connected graphs

A graph G = (V, E) is called weakly four-connected if G is 4-edge-connected and G − x is 2-edge-connected for all x ∈ V . We give sufficient conditions for the existence of ‘splittable’ vertices of degree four in weakly four-connected graphs. By using these results we prove that every minimally weakly fourconnected graph on at least four vertices contains at least three ‘splittable’ vertices of degree four, which gives rise to an inductive construction of weakly four-connected graphs. Our results can also be applied in the problem of finding 2-connected orientations of graphs.