Finite rigid subgraphs of pants graphs

Let S g , n be an orientable surface of genus g with n punctures. We identify a finite rigid subgraph X g , n of the pants graph P ( S g , n ) , that is, a subgraph with the property that any simplicial embedding of X g , n into any pants graph P ( S g ′ , n ′ ) is induced by an embedding S g , n → S g ′ , n ′ . This extends results of the third author for the case of genus zero surfaces.