Concatenation of Graphs

An operation of concatenation is introduced for graphs. Then strings are viewed as expressions denoting graphs, and string languages are interpreted as graph languages. For a class K of string languages, Int(K) is the class of all graph languages that are interpretations of languages from K. For the class REG of regular languages, Int(REG) might be called the class of regular graph languages; it equals the class of graph languages generated by linear Hyperedge Replacement Systems. Two characterizations are given of the largest class K such that Int(K) = Int(K). Context-free graph languages are generated by context-free graph grammars, which are graph replacement systems. One of the most popular types of context-free graph grammar is the Hyperedge Replacement System, or HR grammar (see, e.g., [9]). A completely different way of generating graphs (introduced in [1]) is to select a number of graph operations, to generate a set of expressions (built from these operations), and to interpret the expressions as graphs. The set of expressions is generated by a classical context-free grammar generating strings (or a regular tree grammar). It is shown in [1] that, for a particular collection of graph operations, this new way of generating graphs is equivalent with the HR grammar. Other work on the generation of graphs through graph expressions is in, e.g., [2, 3, 4, 5, 11]. We introduce a new, natural operation on graphs (which is a simple variation of the graph operations in [1]). Due to its similarity to concatenation of strings, it is called concatenation of graphs. Together with the sum operation of graphs (as defined in [1]) and all constant graphs, a collection of graph operations is obtained that is simpler than the one in [1], but also has the power of the HR grammar (which is our first result). Let us be a bit more precise. We consider the multi-pointed graphs (or multi-pointed hypergraphs) of [9]. For simplicity we will restrict ourselves in this paper to graphs, but all arguments also hold for hypergraphs. A multi-pointed graph is a directed, edgelabeled graph g with a designated sequence begin(g) of “begin nodes” and a designated