Analytic and topological nets

Abstract

We characterize which planar graphs arise as the pullback, under a rational map r, of an analytic Jordan curve passing through the critical values of r. We also prove that such pullbacks are dense within the collection of $f^{-1}\left(\Sigma \right)$, where f is a branched cover of the sphere and Σ is a Jordan curve passing through the branched values of f.