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The set of factorizations of permutations in to $m$ transpositions of somesymmetric group $\mathcal{S}_n$ is naturally in bijection with the set ofgraphs of order $n$ and size $m$ with both edges and vertices labeled. Wedefine a notion of duality (the \emph{mind-body duality}) for factorizationsand such labeled graphs and interpret it in terms of Properly Embedded Graphs,a class of graphs embedded in a bounded compact oriented surface with all thevertices lying in the boundary, and show a close connection of this dualitywith the Hurwitz action of the Braid Group. Connections with the theory ofCellularly Embedded Graphs are highlighted and hints of possible applicationsare given. In this paper we focus on developing the necessary theory, leavingspecific applications and further developments for future projects.

A duality for labeled graphs and factorizations with applications to graph embeddings and Hurwitz enumeration