On outward and inward productions in the categorical graph-grammar approach and ∆-grammars

We consider the relationship between three ways of defining graph derivability. That the traditional double-pushout approach and Banach’s inward version are equivalent in the case of injective left-hand side is proved in a purely categorical setting. In the case of noninjective left-hand sides, equivalence can be shown in special categories if the righthand side is injective. Both approaches have the same generative power in the category of graphs if the pushout connecting the outward production with the inward one is a pullback as well. Finally, it is shown that Banach’s point of view establishes a close relationship between the categorical approach and Kaplan’s ∆-grammars allowing a slight generalization of ∆-grammars and making them an operational description of the categorical approach.