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I owe special thanks to my advisor Hannah Markwig for many helpful conversations, as this paper grew out of the preparation of my Master's thesis. Moreover, I want to thank Norman Do for an interesting discussion about his paper \cite{DN13} and an anonymous referee for many helpful remarks on an earlier version of this paper. Thanks are also due to Frank-Olaf Schreyer and Roland Speicher. Finally, I would like to thank Malte Grajewski for his help with Figure \ref{firstcase}.  The author gratefully acknowledges partial support by DFG SFB-TRR 195 "Symbolic tool in mathematics and their applications", project A 14 "Random matrices and Hurwitz numbers" (INST 248/238-1).
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We can compute Hurwitz numbers in terms of isomorphism classes of branching graphs of type $(g,\mu,\nu)$. We denote the set of all isomorphism classes of branching graphs of type $(g,\mu,\nu)$ by $\mathcal{B}_g(\mu,\nu)$ .
  \item Relabel the edges of $\tilde{F}$ by $\mathcal{E}-E$, such that the edge labeled $i$ is relabeled by the $i-th$ element of $\mathcal{E}-E$ in the natural order. Add the half-edges in $\{1,\dots ,m\}-(\mathcal{E}-E)$ to the pruned face, such that half-edges are labeled cyclically counterclockwise by $\{1,\dots,m\}$.
