Consider a branched self-cover of the sphere: a finite set $P$ in $S^2$, and a map $f: (S^2,P) \to (S^2, P)$ which is a covering away from~$P$. When is $f$ equivalent to a rational map? W. Thurston gave one answer in 1982, based on the non-existence of invariant multi-curves. In this talk, we will propose another answer, based on metric graphs satisfying a self-embedding condition.

The question is analogous to the question of when a surface automorphism is pseudo-Anosov, and the two answers are analogous to two characterizations of pseudo-Anosov diffeomorphisms, as either not having a reducing system, or as having a measured foliation invariant up to scale.