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90
On the fixed parameter complexity of graph enumeration problems definable in monadic secondorder logic
, 2001
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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 54 (6 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.
Locally excluding a minor
, 2007
"... We introduce the concept of locally excluded minors. Graph classes locally excluding a minor generalise the concept of excluded minor classes but also of graph classes with bounded local treewidth and graph classes with bounded expansion. We show that firstorder modelchecking is fixedparameter t ..."
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Cited by 31 (12 self)
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We introduce the concept of locally excluded minors. Graph classes locally excluding a minor generalise the concept of excluded minor classes but also of graph classes with bounded local treewidth and graph classes with bounded expansion. We show that firstorder modelchecking is fixedparameter tractable on any class of graphs locally excluding a minor. This strictly generalises analogous results by Flum and Grohe on excluded minor classes and Frick and Grohe on classes with bounded local treewidth. As an important consequence of the proof we obtain fixedparameter algorithms for problems such as dominating or independent set on graph classes excluding a minor, where now the parameter is the size of the dominating set and the excluded minor. We also study graph classes with excluded minors, where the minor may grow slowly with the size of the graphs and show that again, firstorder modelchecking is fixedparameter tractable on any such class of graphs.
Finding branchdecompositions and rankdecompositions
, 2007
"... Abstract. We present a new algorithm that can output the rankdecomposition of width at most k of a graph if such exists. For that we use an algorithm that, for an input matroid represented over a fixed finite field, outputs its branchdecomposition of width at most k if such exists. This algorithm w ..."
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Cited by 27 (1 self)
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Abstract. We present a new algorithm that can output the rankdecomposition of width at most k of a graph if such exists. For that we use an algorithm that, for an input matroid represented over a fixed finite field, outputs its branchdecomposition of width at most k if such exists. This algorithm works also for partitioned matroids. Both these algorithms are fixedparameter tractable, that is, they run in time O(n 3) for each fixed value of k where n is the number of vertices / elements of the input. (The previous best algorithm for construction of a branchdecomposition or a rankdecomposition of optimal width due to Oum and Seymour [Testing branchwidth. J. Combin. Theory Ser. B, 97(3) (2007) 385–393] is not fixedparameter tractable.)
How to solve NPhard graph problems on cliquewidth bounded graphs in polynomial time
 PROCEEDINGS OF THE 27TH WORKSHOP ON GRAPHTHEORETIC CONCEPTS IN COMPUTER SCIENCE (WG 2001), LNCS 2204
, 2001
"... We show that many nonMSO1 NPhard graph problems can be solved in polynomial time on cliquewidth and NLCwidth bounded graphs using a very general and simple scheme. Our examples are partition into cliques, partition into triangles, partition into complete bipartite subgraphs, partition into perfe ..."
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Cited by 23 (3 self)
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We show that many nonMSO1 NPhard graph problems can be solved in polynomial time on cliquewidth and NLCwidth bounded graphs using a very general and simple scheme. Our examples are partition into cliques, partition into triangles, partition into complete bipartite subgraphs, partition into perfect matchings, partition into forests, cubic subgraph, Hamiltonian path, minimum maximal matching, and vertex/edge separation problems.
Width parameters beyond treewidth and their applications
 Computer Journal
, 2007
"... Besides the very successful concept of treewidth (see [Bodlaender, H. and Koster, A. (2007) Combinatorial optimisation on graphs of bounded treewidth. These are special issues on Parameterized Complexity]), many concepts and parameters measuring the similarity or dissimilarity of structures compare ..."
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Cited by 18 (0 self)
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Besides the very successful concept of treewidth (see [Bodlaender, H. and Koster, A. (2007) Combinatorial optimisation on graphs of bounded treewidth. These are special issues on Parameterized Complexity]), many concepts and parameters measuring the similarity or dissimilarity of structures compared to trees have been born and studied over the past years. These concepts and parameters have proved to be useful tools in many applications, especially in the design of efficient algorithms. Our presented novel look at the contemporary developments of these ‘width ’ parameters in combinatorial structures delivers—besides traditional treewidth and derived dynamic programming schemes—also a number of other useful parameters like branchwidth, rankwidth (cliquewidth) or hypertreewidth. In this contribution, we demonstrate how ‘width ’ parameters of graphs and generalized structures (such as matroids or hypergraphs), can be used to improve the design of parameterized algorithms and the structural analysis in other applications on an abstract level.
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 15 (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.
The TreeWidth of CliqueWidth Bounded Graphs Without K n,n
 In Proceedings of GraphTheoretical Concepts in Computer Science, volume 1938 of LNCS
, 2000
"... . We proof that every graph of cliquewidth k which does not contain the complete bipartite graph Kn;n for some n > 1 as a subgraph has treewidth at most 3k(n 1) 1. This immediately implies that a set of graphs of bounded cliquewidth has bounded treewidth if it is uniformly lsparse, closed u ..."
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Cited by 15 (3 self)
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. We proof that every graph of cliquewidth k which does not contain the complete bipartite graph Kn;n for some n > 1 as a subgraph has treewidth at most 3k(n 1) 1. This immediately implies that a set of graphs of bounded cliquewidth has bounded treewidth if it is uniformly lsparse, closed under subgraphs, of bounded degree, or planar. 1 Introduction The cliquewidth of a graph is dened by composition mechanisms for vertexlabeled graphs, see [CO00]. The operations are the vertex disjoint union of labeled graphs, the addition of edges between vertices controlled by some label pair, and a relabeling of the vertices. The used number of labels corresponds to the cliquewidth of the dened graph. Cliquewidth bounded graphs are especially interesting from an algorithmic point of view. A lot of NPcomplete graph problems can be solved in polynomial time for graphs of bounded cliquewidth if the composition tree of the graphs is explicitly given. For example, the set of all graph p...
The recognizability of sets of graphs is a robust property
"... Once the set of finite graphs is equipped with an algebra structure (arising from the definition of operations that generalize the concatenation of words), one can define the notion of a recognizable set of graphs in terms of finite congruences. Applications to the construction of efficient algorith ..."
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Cited by 13 (9 self)
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Once the set of finite graphs is equipped with an algebra structure (arising from the definition of operations that generalize the concatenation of words), one can define the notion of a recognizable set of graphs in terms of finite congruences. Applications to the construction of efficient algorithms and to the theory of contextfree sets of graphs follow naturally. The class of recognizable sets depends on the signature of graph operations. We consider three signatures related respectively to Hyperedge Replacement (HR) contextfree graph grammars, to Vertex Replacement (VR) contextfree graph grammars, and to modular decompositions of graphs. We compare the corresponding classes of recognizable sets. We show that they are robust in the sense that many variants of each signature (where in particular operations are defined by quantifierfree formulas, a quite flexible framework) yield the same notions of recognizability. We prove that for graphs without large complete bipartite subgraphs, HRrecognizability and VRrecognizability coincide. The same combinatorial condition equates HRcontextfree and VRcontextfree sets of graphs. Inasmuch as possible, results are formulated in the more general framework of relational structures. 1
Algorithmic MetaTheorems
 In M. Grohe and R. Neidermeier eds, International Workshop on Parameterized and Exact Computation (IWPEC), volume 5018 of LNCS
, 2008
"... Algorithmic metatheorems are algorithmic results that apply to a whole range of problems, instead of addressing just one specific problem. This kind of theorems are often stated relative to a certain class of graphs, so the general form of a meta theorem reads “every problem in a certain class C of ..."
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Cited by 13 (2 self)
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Algorithmic metatheorems are algorithmic results that apply to a whole range of problems, instead of addressing just one specific problem. This kind of theorems are often stated relative to a certain class of graphs, so the general form of a meta theorem reads “every problem in a certain class C of problems can be solved efficiently on every graph satisfying a certain property P”. A particularly well known example of a metatheorem is Courcelle’s theorem that every decision problem definable in monadic secondorder logic (MSO) can be decided in linear time on any class of graphs of bounded treewidth [1]. The class C of problems can be defined in a number of different ways. One option is to state combinatorial or algorithmic criteria of problems in C. For instance, Demaine, Hajiaghayi and Kawarabayashi [5] showed that every minimisation problem that can be solved efficiently on graph classes of bounded treewidth and for which approximate solutions can be computed efficiently from solutions of certain subinstances, have a PTAS on any class of graphs excluding a fixed minor. While this gives a strong unifying explanation for PTAS of many