B. COURCELLE, J.A. MAKOWSKY, U. ROTICS, Linear time solvable optimization problems on graphs of bounded clique-width, Theory of Computing Systems 33 (2000) 125-150  Home/Search   Document Details and Download   Summary   Related Articles   Check

....much attention. This concept was introduced in connection with graph grammars by Courcelle, Engelfriet and Rozenberg in [10] it is closely related to modular decomposition and extends the notion of treewidth since bounded treewidth implies bounded clique width but not vice versa [12] In [11], Courcelle, Makowsky and Rotics have shown that every graph problem definable in LinEMSOL(# 1,L ) a variant of Monadic Second Order Logic) is linear time solvable on graph classes with bounded clique width if a k expression describing the input graph is given. The problems Vertex Cover, Maximum ....

B. Courcelle, J.A. Makowsky, U. Rotics, Linear time solvable optimization problems on graphs of bounded clique width, Theory of Computing Systems 33 (2000) 125-150

....Obviously, the clique width of graphs having n vertices is at most n, and the clique width of cographs is at most two. A k expression for a graph G of clique width k describes the recursive generation of G by repeatedly applying these operations using at most k di erent labels. 7 Proposition 1 ([10, 11]) Assume that G is not a cograph. The clique width of G is the maximum of the clique width of its prime subgraphs, and the clique width of the complement graph G is at most twice the clique width of G. Recently, the concept of clique width of a graph attracted much attention since it gives a uni ....

....the clique width of G. Recently, the concept of clique width of a graph attracted much attention since it gives a uni ed approach to the ecient solution of many problems on graph classes of bounded clique width via the expressibility of the problems in terms of certain logical expressions (see [10]) Roughly speaking, MSOL( 1 ) is Monadic Second Order Logic with quanti cation over subsets of vertices but not of edges; MSOL( 1;L ) is the extension of MSOL( 1 ) with the addition of labels added to the vertices. LinEMSOL( 1;L ) is the extension of MSOL( 1;L ) which allows to search for ....

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B. Courcelle, J.A. Makowsky, U. Rotics, Linear time solvable optimization problems on graphs of bounded clique width, Theory of Computing Systems 33 (2000) 125-150

.... second order logic become polynomial time solvable when restricted to graphs of bounded tree width [6, 14] and all problems expressible in monadic second order logic using quanti ers on vertices but not on edges become polynomial time solvable when restricted to graphs of bounded clique width [9, 10]. This includes, for example, maximum clique, independent set, minimum dominating or independent dominating set problems as well as k colorability (for xed k) maximum induced matching, induced path, etc. Furthermore, graphs of bounded band width are easily seen to be of bounded tree and ....

B. Courcelle, J.A. Makowsky and U. Rotics, Linear time solvable optimization problems on graphs of bounded clique-width, Theory Comput. Systems 33 (2000), 125-150.

....a connected K 1;3 free bipartite graph, then cwd( e G) 5. A conclusion from Lemmas 1 3 and Decomposition Theorem is as follows. Theorem 1. The clique width of bipartite graphs without a skew star is at most five. For a class of graphs with clique width at most k, Courcelle et al. present in [7] a number of optimization problems, which, given a graph G in the class and an O(f(jV Gj; jV Ej) algorithm to construct a k expression defining G, can be solved for G in time O(f(jV Gj; jV Ej) It has been observed in [14] that canonical decomposition can be realized for a bipartite graph with n ....

....[14] that canonical decomposition can be realized for a bipartite graph with n vertices and m edges in time O(n 2 nm) Obviously, this time is sufficient to find a maximal prime induced subgraph of a bipartite graph. Summarizing, we obtain from the above arguments, Theorem 1 and the results of [7] polynomial algorithms for a Page 16 RRR 20 2001 number of problems which are NP complete in general bipartite graphs (see [15] for a formal definition of these problems) Corollary 1. Given a bipartite graph without a skew star G with n vertices and m edges, one can solve the following problem ....

B. Courcelle, J.A. Makowsky and U. Rotics, Linear time solvable optimization problems on graphs of bounded clique-width, Theory Comput. Systems 33 (2000) 125-150.

....NLC width if an expression for the graph is explicitly given. For example, all graph properties which are expressible in monadic second order logic with quanti cations over vertices and vertex sets (MSO 1 logic) are decidable in linear time on cliquewidth and NLC width bounded graphs, see [CMR98] The MSO 1 logic has been extended by counting mechanisms which allow the expressibility of optimization problems concerning maximal or minimal vertex sets, see [CMR98] All these graph problems expressible in extended MSO 1 logic can also be solved in linear time on clique width and NLC width ....

.... over vertices and vertex sets (MSO 1 logic) are decidable in linear time on cliquewidth and NLC width bounded graphs, see [CMR98] The MSO 1 logic has been extended by counting mechanisms which allow the expressibility of optimization problems concerning maximal or minimal vertex sets, see [CMR98] All these graph problems expressible in extended MSO 1 logic can also be solved in linear time on clique width and NLC width bounded graphs. The work of the second author was supported by the German Research Association (DFG) grant WA 674 9 1. If a graph G has clique width or NLC width ....

B. Courcelle, J.A. Makowsky, and U. Rotics. Linear time solvable optimization problems on graphs of bounded clique width, extended abstract. In Proceedings of Graph-Theoretical Concepts in Computer Science, volume 1517 of LNCS, pages 1-16. Springer-Verlag, 1998.

....NLC width if an expression for the graph is explicitly given. For example, all graph properties which are expressible in monadic second order logic with quanti cations over vertices and vertex sets (MSO 1 logic) are decidable in linear time on cliquewidth and NLC width bounded graphs, see [CMR00] The MSO 1 logic has been extended by counting mechanisms which allow the expressibility of optimization problems concerning maximal or minimal vertex sets, see [CMR00] All these graph problems expressible in extended MSO 1 logic can also be solved in linear time on clique width and NLC width ....

.... over vertices and vertex sets (MSO 1 logic) are decidable in linear time on cliquewidth and NLC width bounded graphs, see [CMR00] The MSO 1 logic has been extended by counting mechanisms which allow the expressibility of optimization problems concerning maximal or minimal vertex sets, see [CMR00] All these graph problems expressible in extended MSO 1 logic can also be solved in linear time on clique width and NLC width bounded graphs. The work of the second author was supported by the German Research Association (DFG) grant WA 674 9 1. If a graph G has clique width or NLC width ....

B. Courcelle, J.A. Makowsky, and U. Rotics. Linear time solvable optimization problems on graphs of bounded clique width. Theory of Computing Systems, 33(2):125-150, 2000.

....of labels, assuming that the graph is given in such a decomposed form. For example, decision, optimization, and enumeration problems expressible in MS1 logic, such as 3 Colorability, MaxClique and #MaxClique, can be solved in linear time on graphs given as clique decompositions of width at most k [4, 5]. And P recognizable problems, such as Hamiltonian Circuit (which is not MS1 expressible [4] can be solved in polynomial time on graphs given as NLC decompositions of width at most k [15] Note here that in theory it does not really matter which decomposition we have. For the transformations ....

B. Courcelle, J. A. Makowsky and U. Rotics, "Linear time solvable optimization problems on graphs of bounded clique width," in Proc. 24th Int. Workshop on Graph-Theoretic Concepts in Computer Science, Lecture Notes in Computer Science 1517 (Springer, Berlin, 1998) pp. 1--16. 22

....tw(G) tw(I(G) and not only the trivial tw(I(G) tw(G) tw(G) 1 2 ) Thus Theorem 4 is valid for both versions of MSO. Arnborg, Lagergren and Seese [ALS91] proved extensions of Courcelle s Theorem for MSO de nable counting and optimization problems. Courcelle, Makowsky, and Rotics [CMR98] consider another, more liberal parameter of graphs called clique width. They prove that if a graph comes with a clique decomposition (the analogue of a tree decomposition for clique width) of bounded width, then MSO model checking is still possible in polynomial time. The problem with this ....

B. Courcelle, J.A. Makowsky, and U. Rotics. Linear time solvable optimization problems on graphs of bounded clique width. In Graph Theoretic Concepts in Computer Science, WG'98, volume 1517 of Lecture Notes in Computer Science, pages 1-16. Springer-Verlag, 1998.

.... and Rozenberg have de ned the class of graphs of clique width at most k as the class of graphs which can be de ned by expressions based on graph operations which use k vertex labels (i.e. k expressions) Given a graph G of bounded clique width, say k, Courcelle, Makowsky and Rotics have shown in [2] that there are linear time solutions for a number of optimization problems on G once a k expression de ning G is given. Let abcd be a P 4 (i.e. a chord less path on the 4 vertices a; b; c; d) of a graph G = V; E) a partner of abcd in G is a vertex x of V fa; b; c; dg such that the set fx; a; ....

....the set fx; a; b; c; dg induces at least two P 4 s. Observe that P 4 sparse graphs de ned by Ho ang in [9] are precisely the graphs where no induced P 4 has a partner, while the graphs having no induced P 4 with more than one partner are de ned to be the P 4 tidy graphs in [7] The authors of [2] have shown that P 4 tidy graphs are of bounded clique width by using the modular decomposition of these graphs. Preprint submitted to Elsevier Preprint 25 August 2000 Roussel, Rusu and Thuillier have introduced the partner limited graphs (or PL graphs) in [10] By de nition no induced P 4 of ....

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B. Courcelle, J.A. Makowsky and U. Rotics. Linear time solvable optimization problems on graphs of bounded clique width (extended abstract). Lecture Notes in Computer Science, 1517:1-16, 1998.

....clique width if the composition tree of the graphs is explicitly given. For example, the set of all graph properties which are expressible in monadic second order logic with quanti cations over vertices and vertex sets (MSO 1 logic) can be solved in linear time on clique width bounded graphs [CMR98] The MSO 1 logic has been extended by counting mechanisms which allow the expressibility of optimization problems, see [CMR98] All these problems expressible in the extended MSO 1 logic can be solved in polynomial time on clique width bounded graphs. Furthermore, a lot of NP complete graph ....

.... expressible in monadic second order logic with quanti cations over vertices and vertex sets (MSO 1 logic) can be solved in linear time on clique width bounded graphs [CMR98] The MSO 1 logic has been extended by counting mechanisms which allow the expressibility of optimization problems, see [CMR98] All these problems expressible in the extended MSO 1 logic can be solved in polynomial time on clique width bounded graphs. Furthermore, a lot of NP complete graph problems which are not expressible in MSO 1 logic or extended MSO 1 logic like Hamiltonicity and the simple max cut problem can ....

B. Courcelle, J.A. Makowsky, and U. Rotics. Linear time solvable optimization problems on graphs of bounded clique width, extended abstract. In Proceedings of Graph-Theoretical Concepts in Computer Science, volume 1517 of LNCS, pages 1-16. Springer-Verlag, 1998.

....decision, optimization and counting problems over graphs in val (T (Bn ) are solvable in polynomial time in the size of the terms of T (Bn ) serving as representations of the graphs. Hence, Q3 and Q5 have positive answers for Bn . A di erent proof for Q3 and generalizations are given in [Rot98,CMR00a,CMR00b] Theorem 3 ( CE95,Cou92,EvO97] If a class of graphs in Gn is tree like, then it is included in val (T (Bm ) for some m n. This gives an intrinsic characterization of the hierarchy de ned by clique width and proves the robustness of the notion. The parsing problem Q4 is known to be ....

B. Courcelle, J.A. Makowsky, and U. Rotics. Linear time solvable optimization problems on graphs of bounded clique-width. Theory of Computing Systems, xx:xx{yy, 2000.

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Bruno Courcelle, Johann A. Makowsky, and Udi Rotics, Linear time solvable optimization problems on graphs of bounded clique-width, Theory Comput. Syst. 33 (2000), no. 2, 125--150.

....The cliquewidth of a graph measures to what degree a graph is similar to an overlapping family of cliques. It was first introduced by Courcelle, Engelfriet and Rozenberg, CER93] We shall not need the technical definition of cliquewidth, and the interested reader is referred to, say, MR99,CO00,CMR00,CMR01] We collect here a few facts to allow comparison of our results on treewidth and possible extensions for cliquewidth. Fact 3 Cliques have cliquewidth 2. Golumbic and Rotics proved in [GR01] Proposition 4 The grids Grid n have cliquewidth n 1. From this, together with Fact 3 we see ....

....bounded treewidth. Then K is MSOL polynomial. In other words, the monadic quantifier hierarchy K collapses to the lowest level of the polynomial hierarchy, i.e. P. The same holds for classes K of cliquewidth at most 2, which includes the cliques, and therefore may have unbounded treewidth, cf. CMR00] 6 . This shows that the converse of theorem 10 does not hold. 1.5 Main results In this paper we discuss under which additional assumption on K a converse of Theorem 10 does hold. This question was first discussed in [Rot98] We analyze the four closure conditions M closed, T closed, S closed ....

B. Courcelle, J.A. Makowsky, and U. Rotics. linear time solvable optimization problems on graphs of bounded clique-width. Theory of Computing Systems, 33.2:125--150, 2000.

....Noble, cf. And98,Nob98] have shown that the Tutte polynomial is computable in polynomial time on graphs of fixed bounded tree width. In [Mak00] this result is extended to the colored Tutte polynomials building on previous work by Arnborg, Courcelle, Lagergren, Makowsky, Rotics and Seese [ALS91,CMR00a,CMR00b]. Our proof had two ingredients, one logical and one combinatorial. The first consists in establishing that the polynomial in question can be expressed in a certain formalism of sums of products where the summation ranges over subsets of edges satisfying a condition in Monadic Second Order Logic ....

B. Courcelle, J.A. Makowsky, and U. Rotics. linear time solvable optimization problems on graphs of bounded clique-width. Theory of Computing Systems, 33.2:125--150, 2000.

....exploiting the above methods. We therefore omit its proof. 6.3 Getting around the coefficient condition For some of the problems studied above it is possible to avoid the coefficient condition over infinite fields. The ideas are already present in [ALS91] in an automata theoretic framework) and [CMR00a] (in a logical framework) Therefore we just outline how some of these problems can be putted into their framework. The problems we can handle that way have to be optimization problems where the objective function has a linear structure. In the framework of weighted hypergraphs D = V; E; C) ....

.... shows that the special linear structure of the evaluation term (which is a MSOR term according to our definition) allows to compute the minimum (or maximum) of a R structure obtained after applying a parse operation in constant time from the corresponding extremal values on the substructure (see [ALS91,CMR00a]) In particular, we do not have to store too many intermediate results. This ideas can be applied to the POS(A) problem: Theorem 13. For ordered fields F the problem (d; k) Gamma POS(A)F can be solved in linear time in n where the constants of the linear bound depend on k; d: Proof. We ....

B. Courcelle, J.A. Makowsky, and U. Rotics. Linear Time Solvable Optimization Problems on Graphs of Bounded Clique Width. Theory of Computing Systems, vol. 33 No.2, 125--150, 2000.

....The compactness theorem of Thomassen [T] asserts that the tree width of an in nite graph is an integer k i k is the maximum value of the tree width of its nite subgraphs. Clique width is another complexity measure on graphs yielding low degree algorithms for certain hard problems (see [CO] [CMR]) A set of nite graphs having bounded tree width has bounded clique width but not vice versa (every nite clique Kn has clique width 2 and tree width n 1) The notion of clique width is based on the de nition of graphs by nite terms over graph operations that build graphs. These operations ....

B. Courcelle, J. Makowsky, U. Rotics, Linear time solvable optimization problems on graphs of bounded clique-width, Th. Comput. Systems 33 (2000) 125150.

....Such a function is called an MS de nable optimization function. The distance function on a connected graph is of this type. We consider also the function F (x 1 ; x k ) Card f X j X VG ; G (x 1 ; x k ; X) g We call it an MS de nable counting function. See Courcelle et al. [CMR] on linear algorithms for MS de nable optimization and counting functions on graphs of bounded clique width (each linearly structured class of graphs has bounded clique width) 5.1 Monoids for optimization functions We let N1 be the monoid N[ f 1 g with addition as binary operation denoted by , ....

B. Courcelle, J. Makowsky, U. Rotics, Linear time solvable optimization problems on graphs of bounded clique-width, Theory Comput. Systems 33 (2000) 125-150.

....that can be linearly written or handled as trees in algorithms. Second, because they yield linear algorithms for the veri cation of MS properties (even NP complete, or on arbitrary levels of the polynomial hierarchy) and even for solving MS de nable optimization or counting problems (see (Courcelle et al. 2000, DAM) A veri cation problem consists in deciding whether a certain property holds or not in a given graph or structure; an optimization problem consists in nding, typically, the largest set of edges or vertices satisfying a property ; a counting problem consists in counting the number of such ....

....algorithm taking as input structures that are not given by their syntax trees. However in concrete cases structures may be structured naturally, just because of the concrete situations they come from. Some structural properties may also help, this is the case for P 4 sparse graphs considered in (Courcelle et al. 2000, ToCS) It may happen that Question 3 has a yes answer whereas the answer to Question 2 is no. For a simple example consider a nite complete binary tree de ned as the value of a term f(f(f( f(a) where f(x) stands for g(x; x) and a stands for itself. Every MS property of the binary tree ....

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B. Courcelle, J.A. Makowsky, and U. Rotics (2000) Linear time solvable optimization problems on graphs of bounded clique-width. Theory of Computing Systems, 33, 125-150.

....that no such expression exists) is known to be polynomial for n at most 3. Its complexity is in NP, but is not known exactly for larger values n . However, in many cases, graphs are given with some hierarchical structuring from which an expression in T (Bn ) can be obtained eciently. See [CMR00a] 3. Classes of graphs of bounded clique width Cographs (they include cliques and bicliques) are the graphs of clique width at most 2. Trees have clique width at most 3. Certain classes of graphs with few induced paths with 4 vertices have clique width at most 4 or 5. A graph of treewidth k ....

....Cographs (they include cliques and bicliques) are the graphs of clique width at most 2. Trees have clique width at most 3. Certain classes of graphs with few induced paths with 4 vertices have clique width at most 4 or 5. A graph of treewidth k has clique width O(2 k ) See [CO98] CMR00a] Theorem 2.2 is helpful to prove that a class has bounded clique width but does not give a good upperbound. For every class of graphs such that an expression in T (Bn ) denoting a given graph in the class can be constructed in polynomial time, the MS de nable decision, optimization, and ....

B. Courcelle, J.A. Makowsky, and U. Rotics. Linear time solvable optimization problems on graphs of bounded clique-width. Theory of Computing Systems, xx:xx{yy, 2000.

....that can be linearly written or handled as trees in algorithms. Second, because they yield linear algorithms for the verification of MS properties (even NP complete, or on arbitrary levels of the polynomial hierarchy) and even for solving MS definable optimization or counting problems (see (Courcelle et al. 2000, DAM) A verification problem consists in deciding whether a certain property holds or not in a given graph or structure; an optimization problem consists in finding, typically, the largest set of edges or vertices satisfying a property ; a counting problem consists in counting the number of such ....

....algorithm taking as input structures that are not given by their syntax trees. However in concrete cases structures may be structured naturally, just because of the concrete situations they come from. Some structural properties may also help, this is the case for P 4 sparse graphs considered in (Courcelle et al. 2000, ToCS) It may happen that Question 3 has a yes answer whereas the answer to Question 2 is no. For a simple example consider a finite complete binary tree defined as the value of a term f(f(f( f(a) where f(x) stands for g(x; x) and a stands for itself. Every MS property of the binary tree ....

[Article contains additional citation context not shown here]

B. Courcelle, J.A. Makowsky, and U. Rotics (2000) Linear time solvable optimization problems on graphs of bounded clique-width. Theory of Computing Systems, 33, 125--150.

....G 2 ; 2 6.4 Transductions and evaluation terms We now want to compute evaluation terms of graphs G = j i;j (G 1 ) and G = ae i j (G 1 ) The operations j i;j and ae i j are special cases of quantifierfree FOL transductions. The following lemma is implicitly already in [2,30] and explicitly in [29]. Lemma 46 Let G; G 1 be graphs over the same universe V , G = j i;j (G 1 ) and let z be an assignment into elements and subsets of V . Let and be formulas in FM;k MS 1 . Then there are formulas j i;j and j i;j ( ae i;j and ae i;j ) in FM;k such that X G;zj= Y 1 ; Y t ....

B. Courcelle, J.A. Makowsky, and U. Rotics. linear time solvable optimization problems on graphs of bounded clique-width. Theory of Computing Systems, 33.2:125--150, 2000.

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B. COURCELLE, J.A. MAKOWSKY, U. ROTICS, Linear time solvable optimization problems on graphs of bounded clique-width, Theory of Computing Systems 33 (2000) 125-150

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B. Courcelle, J. A. Makowsky, and U. Rotics. Linear time solvable optimization problems on graphs of bounded clique width. Theor. Comp. Sc., 33:125--150, 2000.

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B. Courcelle, J. A. Makowsky, and U. Rotics. Linear time solvable optimization problems on graphs of bounded clique width. In Proceedings of the 24th International Workshop on Graph-Theoretic Concepts in Computer Science, number 1517 in Lecture Notes in Computer Science, pages 1--16. Springer-Verlag, 1998.

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B. Courcelle, J. A. Makowsky, and U. Rotics. Linear time solvable optimization problems on graphs of bounded clique width. In Proceedings of the 24th International Workshop on Graph-Theoretic Concepts in Computer Science, number 1517 in Lecture Notes in Computer Science, pages 1--16. Springer-Verlag, 1998.

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B. Courcelle, J.A. Makowsky, U. Rotics, Linear time solvable optimization problems on graphs of bounded clique width, extended abstract in: Conf. Proc. WG'98, LNCS 1517 (1998) 1-16; Theory of Computing Systems 33 (2000) 125-150

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B. Courcelle, J.A. Makowsky, and U. Rotics. Linear time solvable optimization problems on graphs of bounded clique-width. Theory of Computing Systems, 33(2):125--150, 2000.

No context found.

B. Courcelle, J.A. Makowsky, U. Rotics, Linear time solvable optimization problems on graphs of bounded clique width, Theory of Computing Systems 33 (2000) 125-150

No context found.

B. Courcelle, J.A. Makowsky, U. Rotics, Linear time solvable optimization problems on graphs of bounded clique width, extended abstract in: Conf. Proc. WG'98, LNCS 1517 (1998) 1-16; Theory of Computing Systems 33 (2000) 125-150

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B. Courcelle, J.A. Makowsky, U. Rotics, Linear time solvable optimization problems on graphs of bounded clique width, Theory of Computing Systems 33 (2000) 125-150

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B. Courcelle, J.A. Makowsky, U. Rotics, Linear time solvable optimization problems on graphs of bounded clique width, Theory of Computing Systems 33 (2000) 125-150

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B. Courcelle, J.A. Makowsky, U. Rotics, Linear time solvable optimization problems on graphs of bounded clique width, Theory of Computing Systems 33 (2000) 125-150

No context found.

B. Courcelle, J.A. Makowsky, U. Rotics, Linear time solvable optimization problems on graphs of bounded clique-width, extended abstract in: Conf. Proc. WG'98, LNCS 1517 (1998) 1-16; Theory of Computing Systems 33 (2000) 125-150

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B. COURCELLE, J.A. MAKOWSKY, U. ROTICS, Linear time solvable optimization problems on graphs of bounded clique-width, Theory of Computing Systems 33 (2000) 125-150

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Courcelle, B., Makowsky, J. & Rotics, U. 2000, 'Linear time solvable optimization problems on graphs of bounded clique width', Theory Cornput. Systems 33, pp. 125-150.

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Courcelle, B., Makowsky, J. & Rotics, U. 2000, `Linear time solvable optimization problems on graphs of bounded clique width', Theory Comput. Systems 33, pp. 125150.

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