On the classification problem for tetravalent metacirculant graphs

The classification problem for tetravalent metacirculant graphs has been considered first in [4]. For this classification three families of special tetravalent metacirculant graphs, denoted by Phi(1), Phi(2) and Phi(3), have been defined [4]. It has also been proved there that any non-Cayley tetravalent metacirculant graph is isomorphic to a union of finitely many disjoint copies of a non-Cayley graph in one of the families Phi(1), Phi(2) or Phi(3). In this paper we prove that the converse is also true. Further results on non-Cayley graphs in Phi(1), Phi(2) and Phi(3) are also discussed.