Planar Whitehead graphs with cyclic symmetry arising from the study of Dunwoody manifolds

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 we expand its scope by classifying all graphs that satisfy the first five conditions.