A Generative Approach of Graph Relabeling Systems

Graph relabeling systems (GRS's) have been introduced as a suitable tool for coding and proving sequential or distributed algorithms on graphs or networks. These systems do not change the underlying structure of the graph on which they work, but only the labeling of its components (edges or vertices). Each relabeling step is fully determined by the knowledge of a xed size subgraph, the local context of the relabeled occurrence. We introduce an extension of that model allowing the generation of sets of graphs. The so-obtained expanding graph relabeling systems (e-GRS's) are also based on component relabeling and do not use any embedding mechanism. We study the generating power of these systems and prove that, despite their apparent simplicity, they enable us to generate any recursively enumerable set of graphs. We rst show how the \from left to right" natural orientation of a string-graph, that is a graph representation of a string, can be translated by means of vertex labels in such a way that any local transformation of the string can be simulated by a local relabeling of the string-graph vertices. Using this translation, we show that any phrase-structure string grammar can be simulated by an e-GRS. Finally, we provide a way of encoding graphs as strings and an e-GRS, called the decoder, which can convert any string representation of the encoding of a graph into the graph itself.