  A fundamental theorem in the study of Dunwoody manifolds is a classification
of finite graphs on $2n$ vertices that satisfy seven conditions (concerning
planarity, regularity, and a cyclic automorphism of order $n$). Its
significance is that if the presentation complex of a cyclic presentation is a
spine of a 3-manifold then its Whitehead graph satisfies the first five
conditions (the remaining conditions do not necessarily hold). In this paper we
observe that this classification relies implicitly on an unstated, and
unnecessary, 8th condition and that this condition is not necessary for such a
presentation complex to be the spine of a 3-manifold. We expand the scope of
Dunwoody's classification by classifying all graphs that satisfy the first five
conditions.
