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 is an element of 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 four-connected 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. (C) 2006 Wiley Periodicals, Inc.