Cyclically five-connected cubic graphs

A cubic graph G is cyclically 5-connected if G is simple, 3-connected, has at least 10 vertices and for every set F of edges of size at most four, at most one component of G\F contains circuits. We prove that if G and H are cyclically 5-connected cubic graphs and H topologically contains G, then either G and H are isomorphic, or (modulo well-described exceptions) there exists a cyclically 5-connected cubic graph G^′ such that H topologically contains G^′ and G^′ is obtained from G in one of the following two ways. Either G^′ is obtained from G by subdividing two distinct edges of G and joining the two new vertices by an edge, or G^′ is obtained from G by subdividing each edge of a circuit of length five and joining the new vertices by a matching to a new circuit of length five disjoint from G in such a way that the cyclic orders of the two circuits agree. We prove a companion result, where by slightly increasing the connectivity of H we are able to eliminate the second construction. We also prove versions of both of these results when G is almost cyclically 5-connected in the sense that it satisfies the definition except for 4-edge cuts such that one side is a circuit of length four. In this case G^′ is required to be almost cyclically 5-connected and to have fewer circuits of length four than G. In particular, if G has at most one circuit of length four, then G^′ is required to be cyclically 5-connected. However, in this more general setting the operations describing the possible graphs G^′ are more complicated.