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The monadic secondorder logic of graphs I. Recognizable sets of Finite Graphs
 Information and Computation
, 1990
"... The notion of a recognizable sef offinite graphs is introduced. Every set of finite graphs, that is definable in monadic secondorder logic is recognizable, but not vice versa. The monadic secondorder theory of a contextfree set of graphs is decidable. 0 19W Academic Press. Inc. This paper begins ..."
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Cited by 208 (14 self)
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The notion of a recognizable sef offinite graphs is introduced. Every set of finite graphs, that is definable in monadic secondorder logic is recognizable, but not vice versa. The monadic secondorder theory of a contextfree set of graphs is decidable. 0 19W Academic Press. Inc. This paper begins an investigation of the monadic secondorder 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 hyperedgelabelled 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 firstorder logic or in secondorder logic. It turns out that monadic secondorder 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 secondorder 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 secondorder logic) is recognizable. * This work has been supported by the “Programme de Recherches Coordonntes: Mathematiques et Informatique.”
The Expression Of Graph Properties And Graph Transformations In Monadic SecondOrder Logic
, 1997
"... By considering graphs as logical structures, one... ..."
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Cited by 147 (39 self)
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By considering graphs as logical structures, one...
Upper Bounds to the CliqueWidth of Graphs
 Discrete Applied Mathematics
, 1997
"... A graph complexity measure that we call cliquewidth is associated in a natural way with certain graph decompositions, more or less like treewidth is associated with treedecomposition which are, actually, hierarchical decompositions of graphs. In general, a decomposition of a graph G can be viewe ..."
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Cited by 57 (16 self)
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A graph complexity measure that we call cliquewidth is associated in a natural way with certain graph decompositions, more or less like treewidth is associated with treedecomposition 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 NPcomplete problems have linear algorithms on graphs of treewidth or of cliquewidth bounded by some fixed k, and the same will hold for graphs of cliquewidth at most k. The graph operations upon which cliquewidth and the related decomp...
The monadic secondorder logic of graphs XIV: uniformly sparse graphs and edge set quantifications
, 2003
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The monadic secondorder logic of graphs XV: On a Conjecture by D. Seese
 Journal of Applied Logic
, 2006
"... A conjecture by D. Seese states that if a set of graphs has a decidable monadic secondorder theory, then it is the image of a set of trees under a transformation defined by monadic secondorder formulas. We prove that the general case of this conjecture is equivalent to the particular cases of dire ..."
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Cited by 18 (7 self)
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A conjecture by D. Seese states that if a set of graphs has a decidable monadic secondorder theory, then it is the image of a set of trees under a transformation defined by monadic secondorder 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 secondorder formulas. By using a counting argument, we show the intrinsic limits of the methods used so far to handle this conjecture.
On the expression of graph properties in some fragments of monadic secondorder logic
 In Descriptive Complexity and Finite Models: Proceedings of a DIAMCS Workshop
, 1996
"... ABSTRACT: We review the expressibility of some basic graph properties in certain fragments of Monadic SecondOrder logic, like the set of MonadicNP 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 wher ..."
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Cited by 13 (1 self)
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ABSTRACT: We review the expressibility of some basic graph properties in certain fragments of Monadic SecondOrder logic, like the set of MonadicNP 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 3connected, and the other, logically simpler, uses the forbidden Kuratowski subgraphs.
Computing Excluded Minors
"... 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 meth ..."
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Cited by 12 (4 self)
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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 secondorder logic and the decidability of the monadic secondorder 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.
Grid Structures and Undecidable Constraint Theories
 In Proceedings of 6th Colloquium on Trees in Algebra and Programming, volume 1214 of LNCS
, 1999
"... We prove three new undecidability results for computational mechanisms over finite trees: There is a linear, ultrashallow, noetherian and strongly confluent rewrite system R such that the 9 8 fragment of the firstorder theory of onesteprewriting by R is undecidable; the emptiness problem ..."
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Cited by 10 (3 self)
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We prove three new undecidability results for computational mechanisms over finite trees: There is a linear, ultrashallow, noetherian and strongly confluent rewrite system R such that the 9 8 fragment of the firstorder theory of onesteprewriting by R is undecidable; the emptiness problem for tree automata with equality tests between cousins is undecidable; and the 9 8  fragment of the firstorder 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 firstorder terms that represent grids. We extend our representation of grids by terms to a representation of linear twodimensional 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...
The Tree Width of Auxiliary Storage
"... We propose a generalization of results on the decidability of emptiness for several restricted classes of sequential and distributed automata with auxiliary storage (stacks, queues) that have recently been proved. Our generalization relies on reducing emptiness of these automata to finitestate grap ..."
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Cited by 9 (1 self)
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We propose a generalization of results on the decidability of emptiness for several restricted classes of sequential and distributed automata with auxiliary storage (stacks, queues) that have recently been proved. Our generalization relies on reducing emptiness of these automata to finitestate graph automata (without storage) restricted to monadic secondorder (MSO) definable graphs of bounded treewidth, where the graph structure encodes the mechanism provided by the auxiliary storage. Our results outline a uniform mechanism to derive emptiness algorithms for automata, explaining and simplifying several existing results, as well as proving new decidability results. Categories and Subject Descriptors F.1.1 [Theory of Computation]: