This paper deals with rewriting of directed ordered graphs over a graded alphabet G. The author shows that if graphs are represented by vertices and edges direct derivations accordin to \textit{H. Ehrig}, \textit{M. Pfender}, and \textit{H. J. Schneider} [Graph Grammars: an algebraic approach, Proc. 14th Annual Conf. Switching Automata Theory, 167-180 (1973)] can be simulated using a single push-out of partial morphisms. In the following graphs representing terms of the free G-algebra over a given set of variables are considered. Let A, B, C be graphs, \(f: A\to B\) a total function and \(g: A\to C\) be an occurrence, then the push-out of f and g yields a unique graph structure D provided that certain conditions are satisfied. If A, B, C represent terms exp(A), exp(B), exp(C) and f is associated with a term rewriting rule exp(A)\(\to \exp (B)\), then D represents a term exp(D) which one obtains by applying the rule exp(A)\(\to \exp (B)\) to exp(C). Finally the author gives a necessary and sufficient condition for local confluence of a final system of graph rewriting rules associated with term rewriting rules. In the framework of graph grammars according to the Berlin School a similar result holds using the embedding theorem [cf. \textit{H. Ehrig}, Lect. Notes Comput. Sci. 73, 1-69 (1979; Zbl 0407.68072)].