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Linear time solvable optimization problems on graphs of bounded cliquewidth, Extended abstract
 Graph Theoretic Concepts in Computer Science, 24th International Workshop, WG ’98, Lecture Notes in Computer Science
, 1998
"... Abstract. Hierarchical decompositions of graphs are interesting for algorithmic purposes. There are several types of hierarchical decompositions. Tree decompositions are the best known ones. On graphs of treewidth at most k, i.e., that have tree decompositions of width at most k, where k is fixed, ..."
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Cited by 113 (20 self)
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Abstract. Hierarchical decompositions of graphs are interesting for algorithmic purposes. There are several types of hierarchical decompositions. Tree decompositions are the best known ones. On graphs of treewidth at most k, i.e., that have tree decompositions of width at most k, where k is fixed, every decision or optimization problem expressible in monadic secondorder logic has a linear algorithm. We prove that this is also the case for graphs of cliquewidth at most k, where this complexity measure is associated with hierarchical decompositions of another type, and where logical formulas are no longer allowed to use edge set quantifications. We develop applications to several classes of graphs that include cographs and are, like cographs, defined by forbidding subgraphs with “too many ” induced paths with four vertices. 1.
On the fixed parameter complexity of graph enumeration problems definable in monadic secondorder logic
, 2001
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A Spatial Logic for Querying Graphs
 In Proc. of ICALP, volume 2380 of LNCS
, 2001
"... We study a spatial logic for reasoning about labelled directed graphs, and the application of this logic to provide a query language for analysing and manipulating such graphs. We give a graph description using constructs from process algebra. We introduce a spatial logic in order to reason loca ..."
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Cited by 62 (5 self)
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We study a spatial logic for reasoning about labelled directed graphs, and the application of this logic to provide a query language for analysing and manipulating such graphs. We give a graph description using constructs from process algebra. We introduce a spatial logic in order to reason locally about disjoint subgraphs. We extend our logic to provide a query language which preserves the multiset semantics of our graph model. Our approach contrasts with the more traditional setbased semantics found in query languages such as TQL, Strudel and GraphLog.
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...
Approximating cliquewidth and branchwidth
 JOURNAL OF COMBINATORIAL THEORY, SERIES B
, 2006
"... We construct a polynomialtime algorithm to approximate the branchwidth of certain symmetric submodular functions, and give two applications. The first is to graph “cliquewidth”. Cliquewidth is a measure of the difficulty of decomposing a graph in a kind of treestructure, and if a graph has cl ..."
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Cited by 56 (5 self)
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We construct a polynomialtime algorithm to approximate the branchwidth of certain symmetric submodular functions, and give two applications. The first is to graph “cliquewidth”. Cliquewidth is a measure of the difficulty of decomposing a graph in a kind of treestructure, and if a graph has cliquewidth at most k then the corresponding decomposition of the graph is called a “kexpression”. We find (for fixed k) an O(n 9 log n)time algorithm that, with input an nvertex graph, outputs either a (2 3k+2 − 1)expression for the graph, or a true statement that the graph has cliquewidth at least k + 1. (The best earlier algorithm algorithm, by Johansson [13], constructed a k log nexpression for graphs of cliquewidth at most k.) It was already known that several graph problems, NPhard on general graphs, are solvable in polynomial time if the input graph comes equipped with a kexpression (for fixed k). As a consequence of our algorithm, the same conclusion follows under the weaker hypothesis that the input graph has cliquewidth at most k (thus, we no longer need to be provided with an explicit kexpression). Another application is to the area of matroid branchwidth. For fixed k, we find an O(n 4)time algorithm that, with input an nelement matroid in terms of its rank oracle, either outputs a branchdecomposition of width at most 3k − 1 or a true statement that the matroid has branchwidth at least k + 1. The previous algorithm by Hliněn´y [11] was only for representable matroids.
Tutorial introduction to graph transformation: A software engineering perspective
 In Proc. of the First International Conference on Graph Transformation (ICGT 2002
, 2002
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Computing crossing numbers in quadratic time
 J. Comput. Syst. Sci
, 2004
"... We show that for every fixed k ≥ 0 there is a quadratic time algorithm that decides whether a given graph has crossing number at most k and, if this is the case, computes a drawing of the graph in the plane with at most k crossings. 1. ..."
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Cited by 29 (0 self)
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We show that for every fixed k ≥ 0 there is a quadratic time algorithm that decides whether a given graph has crossing number at most k and, if this is the case, computes a drawing of the graph in the plane with at most k crossings. 1.
Monadic SecondOrder Logic, Graph Coverings and Unfoldings of Transition Systems
"... We prove that every monadic secondorder property of the unfolding of a transition system is a monadic secondorder property of the system itself. An unfolding is an instance of the general notion of graph covering. We consider two more instances of this notion. A similar result is possible for ..."
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Cited by 27 (6 self)
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We prove that every monadic secondorder property of the unfolding of a transition system is a monadic secondorder property of the system itself. An unfolding is an instance of the general notion of graph covering. We consider two more instances of this notion. A similar result is possible for one of them but not for the other.
A Comparison of Tree Transductions defined by Monadic Second Order Logic and by Attribute Grammars
, 1998
"... . Two wellknown formalisms for the specication and computation of tree transductions are compared: the mso graph transducer and the attributed tree transducer with lookahead, respectively. The mso graph transducer, restricted to trees, uses monadic second order logic to dene the output tree in ..."
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Cited by 25 (8 self)
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. Two wellknown formalisms for the specication and computation of tree transductions are compared: the mso graph transducer and the attributed tree transducer with lookahead, respectively. The mso graph transducer, restricted to trees, uses monadic second order logic to dene the output tree in terms of the input tree. The attributed tree transducer is an attribute grammar in which all attributes are trees; it is preceded by a lookahead phase in which all attributes have nitely many values. The main result is that these formalisms are equivalent, i.e., that the attributed tree transducer with lookahead is an appropriate implementation model for the tree transductions that are speciable in mso logic. This result holds for mso graph transducers that produce trees with shared subtrees. If no sharing is allowed, the attributed tree transducer satises the single use restriction. 1 Introduction Formulas of monadic second order (mso) logic can be used to express properti...