  We develop a notion of a dual of a graph, generalizing the definition of
Goulden and Yong (which only applied to trees), and reproving their main result
using our new notion. We in fact give three definitions of the dual: a
graph-theoretic one, an algebraic one, and a combinatorial "mind-body" dual,
showing that they are in fact the same, and are also the same (on trees) as the
topological dual developed by Goulden and Yong. Goulden and Yong use their dual
to define a bijection between the vertex labeled trees and the factorizations
of the permutation $(n, \ldots, 1)$ into $n-1$ transpositions, showing that
their bijection has a particular structural property. We reprove their result
using our dual instead.
