Edge-preserving maps of curve graphs

Suppose S1 and S2 are orientable surfaces of finite topological type such that S1 has genus at least 3 and the complexity of S1 is an upper bound of the complexity of S2. Let φ:C(S1)→C(S2) be an edge-preserving map; then S1 is homeomorphic to S2, and in fact φ is induced by a homeomorphism. To prove this, we use several simplicial properties of rigid expansions, which we prove here.