Decidable boundedness problems for sets of graphs generated by hyperedge-replacement

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. Various significant sets of graphs such as the set of series-parallel graphs, the set of (maximum) outerplanar graphs, the set of k-trees, and the set of graphs of cyclic bandwidth ⩽ k can be generated by hyperedge-replacement graph grammars. Hence, the study in this paper is not only attributed to the area of graph grammars but may also interest those who investigate graph-theoretic properties of particular sets of graphs.