Sufficient condition for a topological self-similar set to be a self-similar set

A self-similar set always possesses a self-similar topological structure coded by the shift space (symbolic space), which is considered as the coordinate system for this set. On the contrary, it is known that given a compact set K with self-similar topological structure, there may not exist a metric d such that $(K,d)$ is a self-similar set with the same topological structure. We provide an easy-to-use sufficient condition for the existence of such metric d in terms of the associated graph with respect to the self-similar topological structure. Therefore, one can easily construct a required self-similar set from the shift space by specifying the topological structure.