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.Comment: 47 pages, 5 figures. Revised according to referee's comments. To
appear in J. Combin. Theory Ser.
