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.

中文翻译：

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从 Dunwoody 流形的研究中产生的具有循环对称性的平面怀特海图

Dunwoody 流形研究中的一个基本定理是满足七个条件（关于平面性、正则性和 $n$ 阶循环自同构）的 $2n$ 顶点上的有限图的分类。它的意义在于，如果循环表示的表示复合体是 3 流形的脊，那么它的怀特黑德图满足前五个条件（其余条件不一定成立）。在本文中，我们观察到这种分类隐含地依赖于未说明的、不必要的第 8 个条件，我们通过对满足前五个条件的所有图进行分类来扩展其范围。