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

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