Decidable boundedness problems for hyperedge-replacement graph grammars

Abstract

Consider a class C of hyperedge-replacement graph grammars and a numeric function on graphs like the number of edges, the degree (i.e., the maximum of the degrees of all nodes of a graph), the number of simple paths, the size of a maximum set of independent nodes, etc. Each such function induces a Boundedness Problem for the class C: Given a grammar HRG in C, are the function values of all graphs in the language L(HRG), generated by HRG, bounded by an integer or not? We show that the Boundedness Problem is decidable if the corresponding function is compatible with the derivation process of the grammars in C and if it is composed of maxima, sums, and products in a certain way. This decidability result applies particularly to the examples listed above.