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

## TL;DR

This paper revises Dunwoody's classification of certain planar, cyclically symmetric graphs related to 3-manifold spines, showing that an unnecessary condition can be omitted, thus broadening the classification scope.

## Contribution

It identifies and removes an unnecessary condition from Dunwoody's classification, expanding the set of graphs recognized as relevant to 3-manifold spines.

## Key findings

- The 8th condition is unnecessary for the classification.
- All graphs satisfying the first five conditions are classified.
- The scope of Dunwoody's classification is expanded.

## Abstract

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.