Many-sorted magmas. As in many other works, we use the term magma for what is usually called an algebra. The words "algebra" and "algebraic" are used in many different situations with different meanings. We prefer to avoid them completely and use fresh words. For a set we shall use the term "equational" introduced by Mezei and Wright [25] rather than the term "algebraic" introduced by Eilenberg and Wright Cl81. Many-sorted notions are studied in detail in Ehrig and Mahr [ 171 and Wirsing [ 3 11. We mainly review the notations. We shall use infinite sets of sorts and infinite signatures. For this reason, we need to pay a certain attention to effectivity questions. Let Y be a set called the set of sorts. An Y-signature is a set I; given with two mappings ~1: F+ ,Y* (the urity mapping), and c F-+ Y (the sort mapping). The length of cc(f) is called the rank off, and is denoted by p(f). The profile of ,f in F is the pair (cc(f), a(f)) written sI x s2 x. .. x s,-+ a(f), where CC(~) = (sl,. .. . 3,). An F-magma (i.e., an F-algebra in the sense of [ 173 and [ 311) is an object M = ((M,),, .4r, (fM)EF)), where M, is a nonempty set, for each s in Y, called the domain of sort s of M, and fM is a total mapping: BRUNO COURCELLE M U)-+ Mm for each f~ F.
The study of graph structure has advanced in recent years with great strides: finite graphs can be described algebraically, enabling them to be constructed out of more basic elements. Separately the properties of graphs can be studied in a logical language called monadic second-order logic. In this book, these two features of graph structure are brought together for the first time in a presentation that unifies and synthesizes research over the last 25 years. The authors not only provide a thorough description of the theory, but also detail its applications, on the one hand to the construction of graph algorithms, and, on the other to the extension of formal language theory to finite graphs. Consequently the book will be of interest to graduate students and researchers in graph theory, finite model theory, formal language theory, and complexity theory.
Hierarchical decompositions of graphs are interesting for algorithmic purposes. Many NP complete problems have linear complexity on graphs with tree-decompositions of bounded width. We investigate alternate hierarchical decompositions that apply to wider classes of graphs and still enjoy good algorithmic properties. These decompositions are motivated and inspired by the study of vertex-replacement context-free graph grammars. The complexity measure of graphs associated with these decompositions is called clique width. In this paper we bound the clique width of a graph in terms of its tree width on the one hand, and of the clique width of its edge complement on the other. ?
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