Cubic planar hamiltonian graphs of various types

Let U be the set of cubic planar hamiltonian graphs, A the set of graphs G in U   such that G-vG-v is hamiltonian for any vertex vv of G, B the set of graphs G in U   such that G-eG-e is hamiltonian for any edge e of G, and C the set of graphs G in U such that there is a hamiltonian path between any two different vertices of G  . With the inclusion and/or exclusion of the sets A,BA,B, and C, U is divided into eight subsets. In this paper, we prove that there is an infinite number of graphs in each of the eight subsets.