Abstract: The problem of invariant graphs for rational maps was studied. It was mainly proved that if f is a post-critically finite rational map, whose post-critical set consists of one superattracting fixed point and two points in the Julia set, and whose Julia set is a Sierpinski curve, then an fk-invariant graph containing the post-critical set of f for each sufficiently large k could be found.

Key words: rational maps, post-critically finite, invariant graphs, Thurston map