  Thurston obtained a combinatorial characterization for generic branched
self-coverings that preserve the orientation of the oriented 2-sphere by
associating a planar graph to them [arXiv:1502.04760]. In this work, the
Thurston result is generalized to any branched covering of the oriented
2-sphere. To achieve that the notion of local balance introduced by Thurston is
generalized. As an application, a new proof for a Theorem of
Eremenko-Gabrielov-Mukhin-Tarasov-Varchenko [MR1888795], [MR2552110] is
obtained. This theorem corresponded to a special case of the B. \& M. Shapiro
conjecture. In this case, it refers to generic rational functions stating that
a generic rational function $ R : \mathbb{C}\mathbb{P}^1 \rightarrow
\mathbb{C}\mathbb{P}^1$ with only real critical points can be transformed by
post-composition with an automorphism of $\mathbb{C}\mathbb{P}^1$ into a
quotient of polynomials with real coefficients. Operations against balanced
graphs are introduced.
