  Gene assembly is an intricate biological process that has been studied
formally and modeled through string and graph rewriting systems. Recently, a
restriction of the general (intramolecular) model, called simple gene assembly,
has been introduced. This restriction has subsequently been defined as a string
rewriting system. We show that by extending the notion of overlap graph it is
possible to define a graph rewriting system for two of the three types of rules
that make up simple gene assembly. It turns out that this graph rewriting
system is less involved than its corresponding string rewriting system.
Finally, we give characterizations of the `power' of both types of graph
rewriting rules. Because of the equivalence of these string and graph rewriting
systems, the given characterizations can be carried over to the string
rewriting system.
