Term graph rewriting provides a simple mechanism to

finitely represent restricted forms of infinitary term

rewriting. The correspondence between infinitary 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

infinitary rewriting on the side of term graphs. We aim to fill this

gap by devising two modes of convergence based on a partial order

resp. a metric on term graphs. The thus obtained structures

generalise corresponding modes of convergence that are usually

studied in infinitary 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 resulting infinitary calculi of term graph rewriting

exhibit the same correspondence as we know it from term rewriting:

Convergence via the partial order is a conservative extension of the

metric convergence.