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

The set of factorizations of permutations in to $m$ transpositions of some symmetric group $\mathcal{S}_n$ is naturally in bijection with the set of graphs of order $n$ and size $m$ with both edges and vertices labeled. We define a notion of duality (the \emph{mind-body duality}) for factorizations and 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 the vertices lying in the boundary, and show a close connection of this duality with the Hurwitz action of the Braid Group. Connections with the theory of Cellularly Embedded Graphs are highlighted and hints of possible applications are given. In this paper we focus on developing the necessary theory, leaving specific applications and further developments for future projects.