Term graph rewriting provides a simple mechanism to ¿nitely represent restricted forms of in¿nitary term rewriting. The correspondence between in¿nitary term
rewriting and term graph rewriting has been studied to some extent. However, this endeavour is impaired by the lack of an appropriate counterpart of in¿nitary rewriting on the
side of term graphs. We aim to ¿ll this gap by devising two modes of convergence based
on a partial order respectively a metric on term graphs. The thus obtained structures
generalise corresponding modes of convergence that are usually studied in in¿nitary term
rewriting.
We argue that this yields a common framework in which both term rewriting and term
graph rewriting can be studied. In order to substantiate our claim, we compare convergence
on term graphs and on terms. In particular, we show that the modes of convergence on
term graphs are conservative extensions of the corresponding modes of convergence on
terms and are preserved under unravelling term graphs to terms. Moreover, we show that
many of the properties known from in¿nitary term rewriting are preserved. This includes
the intrinsic completeness of both modes of convergence and the fact that convergence via
the partial order is a conservative extension of the metric convergence.