# Efficient algorithms on context-free graph languages

## Abstract

A number of different graph grammar types have been called ”context-free” in the literature. We consider two recent such formalisms, *boundary node-label controlled* (BNLC) and *hyperedge replacement* (HR) grammars, from a complexity-theoretical point of view. It is shown that all HR languages, the members of which satisfy a certain separability restriction, are contained in LOGCFL, the class of sets which are log-space reducible to context-free (string) languages. In particular, this implies the existence of efficient sequential as well as parallel recognition algorithms for these languages. Since HR grammars can simulate a large class of BNLC grammars, the same holds for an according class of BNLC languages. Thus, in a sense, a large class of BNLC and HR languages are ”close” to context-free string languages.

We then use these results for investigating the complexity of some graph-theoretical problems restricted to HR languages. It is shown that on HR languages which satisfy the abovementioned constraints, a number of problems (some of which are *NP*-complete in the general case) have polynomial-time sequential and very fast and feasible parallel solutions.

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