Finite rigid subgraphs of the pants graphs of punctured spheres

We prove a strong form of finite rigidity for pants graphs of spheres. Specifically, for any n≥4, we construct a finite subgraph Xn of the pants graph P(S0,n) of the n-punctured sphere S0,n with the following property. Any simplicial embedding of Xn into any pants graph P(S0,m) of a punctured sphere is induced by an embedding S0,n→S0,m.