The topological dimension of limits of vertex replacements

Given an initial graph G, one may apply a rule R to G which replaces certain vertices of G with other graphs called replacement graphs to obtain a new graph R(G). By iterating this procedure, a sequence of graphs {Rn(G)} is obtained. When each graph in this sequence is normalized to have diameter one, the resulting sequence may converge in the Gromov–Hausdorff metric. In this paper, we compute the topological dimension of limit spaces of normalized sequences of iterated vertex replacements involving more than one replacement graph. We also give examples of vertex replacement rules that yield fractals.