Abstract not found

We construct a polynomial-time algorithm to approximate the branch-width of certain symmetric submodular functions, and give two applications. The first is to graph “clique-width”. Clique-width is a measure of the difficulty of decomposing a graph in a kind of tree-structure, and if a graph has clique-width at most k then the corresponding decomposition of the graph is called a “k-expression”. We find (for fixed k) an O(n 9 log n)-time algorithm that, with input an n-vertex graph, outputs either a (2 3k+2 − 1)expression for the graph, or a true statement that the graph has clique-width at least k + 1. (The best earlier algorithm algorithm, by Johansson [13], constructed a k log n-expression for graphs of clique-width 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 k-expression (for fixed k). As a consequence of our algorithm, the same conclusion follows under the weaker hypothesis that the input graph has clique-width at most k (thus, we no longer need to be provided with an explicit k-expression). Another application is to the area of matroid branch-width. For fixed k, we find an O(n 4)time algorithm that, with input an n-element matroid in terms of its rank oracle, either outputs a branch-decomposition of width at most 3k − 1 or a true statement that the matroid has branch-width at least k + 1. The previous algorithm by Hliněn´y [11] was only for representable matroids.

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 branch-decomposition of width at most k if such exists. This algorithm works also for partitioned matroids. Both these algorithms are fixed-parameter 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 rank-decomposition of optimal width due to Oum and Seymour [Testing branch-width. J. Combin. Theory Ser. B, 97(3) (2007) 385–393] is not fixed-parameter tractable.)

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 tree-width and graph classes with bounded expansion. We show that first-order model-checking is fixed-parameter 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 tree-width. As an important consequence of the proof we obtain fixed-parameter 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 model-checking is fixed-parameter tractable on any such class of graphs.

We show that many non-MSO1 NP-hard graph problems can be solved in polynomial time on clique-width and NLC-width 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.

Besides the very successful concept of tree-width (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 tree-width and derived dynamic programming schemes—also a number of other useful parameters like branch-width, rank-width (clique-width) or hypertree-width. 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.

A conjecture by D. Seese states that if a set of graphs has a decidable monadic second-order theory, then it is the image of a set of trees under a transformation defined by monadic second-order 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 second-order formulas. By using a counting argument, we show the intrinsic limits of the methods used so far to handle this conjecture.

. We proof that every graph of clique-width k which does not contain the complete bipartite graph Kn;n for some n > 1 as a subgraph has tree-width at most 3k(n 1) 1. This immediately implies that a set of graphs of bounded clique-width has bounded tree-width if it is uniformly l-sparse, closed under subgraphs, of bounded degree, or planar. 1 Introduction The clique-width 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 clique-width of the dened graph. Clique-width bounded graphs are especially interesting from an algorithmic point of view. A lot of NP-complete graph problems can be solved in polynomial time for graphs of bounded clique-width if the composition tree of the graphs is explicitly given. For example, the set of all graph p...

We show that there exists a linear time algorithm for deciding whether a graph of bounded tree-width has clique-width k for some fixed integer k.

The P 4 is the induced path with vertices a; b; c; d and edges ab; bc; cd. The chair (co-P, gem) has a fifth vertex adjacent to b (a and b, a; b; c and d respectively). We give a complete structure description of prime chair-, co-P- and gem-free graphs which implies bounded clique width for this graph class. It is known that this has some nice consequences; very roughly speaking, every problem expressible in a certain kind of Monadic Second Order Logic (quantifying only over vertex set predicates) can be solved efficiently for graphs of bounded clique width. In particular, we obtain linear time for the problems Vertex Cover, Maximum Weight Stable Set (MWS), Maximum Weight Clique, Steiner Tree, Domination and Maximum Induced Matching on chair-, co-P- and gem-free graphs and a slightly larger class of graphs. This drastically improves a recently published O(n ) time bound for Maximum Stable Set on buttery-, chair-, co-P- and gem-free graphs.