The notion of a recognizable sef offinite graphs is introduced. Every set of finite graphs, that is definable in monadic second-order logic is recognizable, but not vice versa. The monadic second-order theory of a context-free set of graphs is decidable. 0 19W Academic Press. Inc. This paper begins an investigation of the monadic second-order logic of graphs and of sets of graphs, using techniques from universal algebra, and the theory of formal languages. (By a graph, we mean a finite directed hyperedge-labelled hypergraph, equipped with a sequence of distinguished vertices.) A survey of this research can be found in Courcelle [ 111. An algebraic structure on the set of graphs (in the above sense) has been proposed by Bauderon and Courcelle [2,7]. The notion of a recognizable set of finite graphs follows, as an instance of the general notion of recognizability introduced by Mezei and Wright in [25]. A graph can also be considered as a logical structure of a certain type. Hence, properties of graphs can be written in first-order logic or in secondorder logic. It turns out that monadic second-order logic, where quantifications over sets of vertices and sets of edges are used, is a reasonably powerful logical language (in which many usual graph properties can be written), for which one can obtain decidability results. These decidability results do not hold for second-order logic, where quantifications over binary relations can also be used. Our main theorem states that every definable set of finite graphs (i.e., every set that is the set of finite graphs satisfying a graph property expressible in monadic second-order logic) is recognizable. * This work has been supported by the “Programme de Recherches Coordonntes: Mathematiques et Informatique.”

By considering graphs as logical structures, one...

A graph complexity measure that we call clique-width is associated in a natural way with certain graph decompositions, more or less like tree-width is associated with tree-decomposition which are, actually, hierarchical decompositions of graphs. In general, a decomposition of a graph G can be viewed as a finite term, written with appropriate operations on graphs, that evaluates to G. Infinitely many operations are necessary to define all graphs. By limiting the operations in terms of some integer parameter k, one obtains complexity measures of graphs. Specifically, a graph G has complexity at most k iff it has a decomposition defined in terms of k operations. Hierarchical graph decompositions are interesting for algorithmic purposes. In fact, many NP-complete problems have linear algorithms on graphs of tree-width or of clique-width bounded by some fixed k, and the same will hold for graphs of clique-width at most k. The graph operations upon which clique-width and the related decomp...

We prove that for any class of finite or countable graphs closed under taking subgraphs, and such that every finite graph in the class has a number of edges linearly bounded in terms of the number of vertices, the edge set quantifications can be replaced by vertex set quantifications in Monadic second-order formulas. This result extends to hypergraphs of bounded rank.

A conjecture by D. Seese states that if a set of graphs has a decidable monadic second-order theory, then it is the image of a set of trees under a transformation defined by monadic second-order formulas. We prove that the general case of this conjecture is equivalent to the particular cases of directed graphs, partial orders and comparability graphs. We present some tools to prove the conjecture for classes of graphs with few cliques or few complete bipartite subgraphs, for line graphs and for interval graphs. We make an essential use of prime graphs, of comparability graphs and of characterizations of graph classes by forbidden induced subgraphs. Our treatment of infinite graphs uses a representation of countable linear orders by binary trees that can be constructed by monadic second-order formulas. By using a counting argument, we show the intrinsic limits of the methods used so far to handle this conjecture.

ABSTRACT: We review the expressibility of some basic graph properties in certain fragments of Monadic Second-Order logic, like the set of Monadic-NP formulas. We focus on cases where a property and its negation are both expressible in the same (or in closely related) fragments. We examine cases where edge quantifications can be eliminated and cases where they cannot. We compare two logical expressions of planarity: one of them is constructive in the sense that it defines a planar embedding of the considered graph if it is planar and 3-connected, and the other, logically simpler, uses the forbidden Kuratowski subgraphs.

We prove three new undecidability results for computational mechanisms over finite trees: There is a linear, ultra-shallow, noetherian and strongly confluent rewrite system R such that the 9 8 -fragment of the first-order theory of one-step-rewriting by R is undecidable; the emptiness problem for tree automata with equality tests between cousins is undecidable; and the 9 8 - fragment of the first-order theory of set constraints with the union operator is undecidable. The common feature of these three computational mechanisms is that they allow us to describe the set of first-order terms that represent grids. We extend our representation of grids by terms to a representation of linear two-dimensional patterns by linear terms, which allows us to transfer classical techniques on the grid to terms and thus to obtain our undecidability results. 1 Introduction The grid structure provides convenient means for encoding computation sequences of Turing machines. A classical encoding...

We introduce the class of tree-interpretable structures which generalises the notion of a prefix-recognisable graph to arbitrary relational structures. We prove that every tree-interpretable structure is finitely axiomatisable in guarded second-order logic with cardinality quantifiers.

By Robertson and Seymour’s graph minor theorem, every minor ideal can be characterised by a finite family of excluded minors. (A minor ideal is a class of graphs closed under taking minors.) We study algorithms for computing excluded minor characterisations of minor ideals. We propose a general method for obtaining such algorithms, which is based on definability in monadic second-order logic and the decidability of the monadic second-order theory of trees. A straightforward application of our method yields algorithms that, for a given k, compute excluded minor characterisations for the minor ideal Tk of all graphs of tree width at most k, the minor ideal Bk of all graphs of branch width at most k, and the minor ideal Gk of all graphs of genus at most k. Our main results are concerned with constructions of new minor ideals from given ones. Answering a question that goes back to Fellows and Langston [11], we prove that there is an algorithm that, given excluded minor characterisations of two minor ideals C and D, computes such a characterisation for the ideal C ∪ D. Furthermore, we obtain an algorithm for computing an excluded minor characterisation for the class of all apex graphs over a minor ideal C, given an excluded minor characterisation for C. (An apex graph over C is a graph G from which one vertex can be removed to obtain a graph in C.) A corollary of this result is a uniform ftpalgorithm for the “distance k from planarity” problem.

We consider a new kind of interpretation over relational structures: finite sets interpretations. Those interpretations are defined by weak monadic second-order (WMSO) formulas with free set variables. They transform a given structure into a structure with a domain consisting of finite sets of elements of the orignal structure. The definition of these interpretations directly implies that they send structures with a decidable WMSO theory to structures with a decidable first-order theory. In this paper, we investigate the expressive power of such interpretations applied to infinite deterministic trees. The results can be used in the study of automatic and tree-automatic structures.