Limits of vertex replacement rules

In an earlier paper [10], J. Previte developed a framework for studying iterated replacements of certain vertices in a graph G by a finite replacement graph H. He showed that the normalized sequence of iterated graphs converges in the Gromov-Hausdorff metric (except for special cases). In this paper, we extend the framework in [10] to iterated vertex replacements where there are at least two replacement graphs and prove a convergence result. We also give examples of vertex replacement rules that yield convergent sequences of graphs.