  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
$f^{-1}(\Sigma)$, where $f$ is a branched cover of the sphere and $\Sigma$ is a
Jordan curve passing through the branched values of $f$.
