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We develop a notion of a dual of a graph, generalizing the definition ofGoulden and Yong (which only applied to trees), and reproving their main resultusing our new notion. We in fact give three definitions of the dual: agraph-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 thetopological dual developed by Goulden and Yong. Goulden and Yong use their dualto define a bijection between the vertex labeled trees and the factorizationsof the permutation $(n, \ldots, 1)$ into $n-1$ transpositions, showing thattheir bijection has a particular structural property. We reprove their resultusing our dual instead.

Three Graph Duals and A Bijection