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160
Linear time solvable optimization problems on graphs of bounded cliquewidth
, 2000
"... 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 dec ..."
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Cited by 168 (22 self)
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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.
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 114 (15 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.
On the fixed parameter complexity of graph enumeration problems definable in monadic secondorder logic
, 2001
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Upper bounds to the clique width of graphs
, 2000
"... Hierarchical decompositions of graphs are interesting for algorithmic purposes. Many NP complete problems have linear complexity on graphs with treedecompositions of bounded width. We investigate alternate hierarchical decompositions that apply to wider classes of graphs and still enjoy good algori ..."
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Cited by 69 (6 self)
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Hierarchical decompositions of graphs are interesting for algorithmic purposes. Many NP complete problems have linear complexity on graphs with treedecompositions 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 vertexreplacement contextfree 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
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 67 (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...
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 67 (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.
Bidimensionality and Kernels
, 2010
"... Bidimensionality theory appears to be a powerful framework in the development of metaalgorithmic techniques. It was introduced by Demaine et al. [J. ACM 2005] as a tool to obtain subexponential time parameterized algorithms for bidimensional problems on Hminor free graphs. Demaine and Hajiaghayi ..."
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Cited by 61 (24 self)
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Bidimensionality theory appears to be a powerful framework in the development of metaalgorithmic techniques. It was introduced by Demaine et al. [J. ACM 2005] as a tool to obtain subexponential time parameterized algorithms for bidimensional problems on Hminor free graphs. Demaine and Hajiaghayi [SODA 2005] extended the theory to obtain polynomial time approximation schemes (PTASs) for bidimensional problems. In this paper, we establish a third metaalgorithmic direction for bidimensionality theory by relating it to the existence of linear kernels for parameterized problems. In parameterized complexity, each problem instance comes with a parameter k and the parameterized problem is said to admit a linear kernel if there is a polynomial time algorithm, called
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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Approximating Rankwidth and Cliquewidth Quickly
, 2006
"... Rankwidth was defined by Oum and Seymour [2006. Approximating cliquewidth and branchwidth. J. Combin. Theory Ser. B 96, 4, 514–528] to investigate cliquewidth. They constructed an algorithm that either outputs a rankdecomposition of width at most f(k) for some function f or confirms that rankw ..."
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Cited by 44 (4 self)
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Rankwidth was defined by Oum and Seymour [2006. Approximating cliquewidth and branchwidth. J. Combin. Theory Ser. B 96, 4, 514–528] to investigate cliquewidth. They constructed an algorithm that either outputs a rankdecomposition of width at most f(k) for some function f or confirms that rankwidth is larger than k in time O(V 9 log V ) for an input graph G = (V,E) and a fixed k. We develop three separate algorithms of this kind with faster running time. We construct an O(V 4)time algorithm with f(k) = 3k + 1 by constructing a subroutine for the previous algorithm; we avoid generic algorithms minimizing submodular functions used by Oum and Seymour. Another one is an O(V 3)time algorithm with f(k) = 24k by giving a reduction from graphs to binary matroids; then we use an approximation algorithm for matroid branchwidth by Hliněny ́ [2005. A parametrized algorithm for matroid branchwidth. SIAM J. Comput. 35, 2, 259–277]. Finally we construct an O(V 3)time algorithm with f(k) = 3k − 1 by combining the ideas of above two cited papers.