In this talk, I will show that for any post-critically finite rational map f, there exists a finite and connected graph G in the Julia set of f, such that f^n(G) ⊂ G for every large n, G contains all the post-critical points in the Julia set and every complementary component of G contains at most one post-critical point. The proof is based on the cluster-exact decomposition of post-critically finite rational maps.

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