In earlier work, we have shown that for confluent term rewrite systems (TRSs for short), innermost polynomial runtime complexity induces polytime computability of the functions defined. In this paper, we generalise this result to full rewriting, for that we exploit graph rewriting. We give a new proof of the adequacy of graph rewriting for full rewriting that allows for a precise control of the resources copied. In sum we completely describe an implementation of rewriting on a Turing machine (TM for short). We show that the runtime complexity of the TRS and the runtime complexity of the TM is polynomially related.