Multiscale Graph Grammars Can Generate Cayley Graphs of Groups and Monoids

Abstract

A graph grammar with parallel replacement of subgraphs, based on the single-pushout approach in graph rewriting, was designed which constructs Cayley graphs of monoids of transformations of a finite set, with permutation groups as a special case. As input, graph-based representations of a finite number of generating transformations have to be specified; they will then correspond to the edge types of the Cayley graph which is the final result of the rewriting process. The grammar has$$7+d$$ rules, where d is the number of generators, and operates at two scale levels. The fine-scale level is the level of elements on which the transformations act and where their composition is calculated by parallel subgraph replacement. The coarse-scale level corresponds to the transformations themselves which are organized in the Cayley graph in a sequential rule application process. Both scale levels are represented in a single graph. The graph grammar was implemented in the programming language XL on the software platform GroIMP, a graph rewriting tool which was originally designed for simulating the growth of plants.