On Balanced Planar Graphs, Following W. Thurston

Let f : S 2 → S 2 be an orientation-preserving branched covering map of degree d ≥ 2, and let Σ be an oriented Jordan curve passing through the critical values of f . Then Γ := f −1 (Σ) is an oriented graph on the sphere. In a group email discussion in Fall 2010, W. Thurston introduced balanced planar graphs and showed that they combinatorially characterize all such Γ, where f has 2d−2 distinct critical values. We give a detailed account of this discussion, along with some examples and an appendix about Hurwitz numbers.