Cycles in quasi 4-connected graphs

A graph G is said to be quasi 4 -connected if G is 3-connected and for each cutset K ~ V( G) with IKI = 3, K is the neighbourhood of a vertex of degree three and G K has precisely two components. It is evident that such graphs need not be 4-connected and yet they exhibit many of the properties of 4-connected graphs. In this paper, we show that, given a set, N ~ E( G) of four independent edges, there is a cycle in G containing N. In fact, we show, more generally: that given a "free edge system" F of size at most 4, in a quasi 4-connected graph G, there is a cycle in G containing F. We also consider the existence of cycles through a given set of vertices and avoiding another set of vertices.