Udi Rotics

On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic

We discuss the parametrized complexity of counting and evaluation problems on graphs where the range of counting is denable in monadic second-order logic (MSOL). We show that for bounded tree-width these problems are solvable in polynomial time. The same holds for bounded clique width in the cases, where the decomposition, which establishes the bound on the clique-width, can be computed in polynomial time and for problems expressible by monadic second-order formulas without edge set quantication. Such quantications are allowed in the case of graphs with bounded tree-width. As applications we discuss in detail how this aects the parametrized complexity of the permanent and the hamiltonian of a matrix, and more generally, various generating functions of MSOL denable graph properties. Finally, our results are also applicable to SAT and ]SAT. ? 2001 Elsevier Science B.V. All rights reserved.

ON THE CLIQUE-WIDTH OF SOME PERFECT GRAPH CLASSES

Graphs of clique–width at most k were introduced by Courcelle, Engelfriet and Rozenberg (1993) as graphs which can be defined by k-expressions based on graph operations which use k vertex labels. In this paper we study the clique–width of perfect graph classes. On one hand, we show that every distance–hereditary graph, has clique–width at most 3, and a 3–expression defining it can be obtained in linear time. On the other hand, we show that the classes of unit interval and permutation graphs are not of bounded clique–width. More precisely, we show that for every

On the Relationship Between Clique-Width and Treewidth

n\in {\mathcal N}

Edge dominating set and colorings on graphs with fixed clique-width

there is a unit interval graph In and a permutation graph Hn having n2 vertices, each of whose clique–width is at least n. These results allow us to see the border within the hierarchy of perfect graphs between classes whose clique–width is bounded and classes whose clique–width is unbounded. Finally we show that every n×n square grid,

Clique-Width is NP-Complete

n\in {\mathcal N}

ON THE CLIQUE–WIDTH OF GRAPH WITH FEW P4'S

, n ≥ 3, has clique–width exactly n+1.

Finding Maximum Induced Matchings in Subclasses of Claw-Free and P5-Free Graphs, and in Graphs with Matching and Induced Matching of Equal Maximum Size

Treewidth is generally regarded as one of the most useful parameterizations of a graphs construction. Clique-width is a similar parameterization that shares one of the powerful properties of treewidth, namely: if a graph is of bounded treewidth (or clique-width), then there is a polynomial time algorithm for any graph problem expressible in monadic second order logic, using quantifiers on vertices (in the case of clique-width you must assume a clique-width parse expression is given). In studying the relationship between treewidth and clique-width, Courcelle and Olariu [Discrete Appl. Math., 101 (2000), pp. 77--114] showed that any graph of bounded treewidth is also of bounded clique-width; in particular, for any graph G with treewidth k, the clique-width of G is at most 4 * 2k - 1 + 1. In this paper, we improve this result by showing that the clique-width of G is at most 3 * 2k - 1 and, more importantly, that there is an exponential lower bound on this relationship. In particular, for any k, there is a graph G with treewidth equal to k, where the clique-width of G is at least

Polynomial-time recognition of clique-width ≤3 graphs

2^{\lfloor k/2\rfloor - 1}

Restrictions of minimum spanner problems

.

Computing graph polynomials on graphs of bounded clique-width

We consider both the vertex and the edge versions of three graph partitioning problems. These problems are dominating set, list-q-coloring with costs (fixed number of colors q) and chromatic number. They are all known to be NP-hard in general. We show that all these problems (except edge-coloring) can be solved in polynomial time on graphs with clique-width bounded by some constant k, if the k-expression of the input graph is also given. In particular, we present the first polynomial algorithms (on these classes) for chromatic number, edge-dominating set and list-q-coloring with costs (fixed number of colors q, both vertex and edge versions). For the two list-q-coloring problems with costs, we even have linear algorithms. Since these classes of graphs include classes like P4-sparse graphs, distance hereditary graphs and graphs with bounded treewidth, our algorithms also apply to these graphs.