The topological dimension of limits of graph substitutions

In 1998, J. Previte developed a framework for studying the dynamics of iterated replacements of certain vertices in a finite graph G by a finite graph H (see [3]). He showed that, except for special cases, the sequence of graphs formed by iterating vertex replacements converges in the Gromov-Hausdorff metric. In this paper we prove that the topological dimension of these limit spaces is one. We also provide examples of graph substitutions whose limit spaces are fractals.