We present an algorithm that constructively produces a solution to the k-dominating set problem for planar graphs in time O(c . To obtain this result, we show that the treewidth of a planar graph with domination number (G) is O( (G)), and that such a tree decomposition can be found in O( (G)n) time. The same technique can be used to show that the k-face cover problem ( find a size k set of faces that cover all vertices of a given plane graph) can be solved in O(c n) time, where c 1 = 3 and k is the size of the face cover set. Similar results can be obtained in the planar case for some variants of k-dominating set, e.g., k-independent dominating set and k-weighted dominating set.

A parameterized problem is xed parameter tractable if it admits a solving algorithm whose running time on input instance (I; k) is f(k) jIj , where f is an arbitrary function depending only on k. Typically, f is some exponential function, e.g., f(k) = c k for constant c. We describe general techniques to obtain growth of the form f(k) = c p k for a large variety of planar graph problems. The key to this type of algorithm is what we call the "Layerwise Separation Property" of a planar graph problem. Problems having this property include planar vertex cover, planar independent set, and planar dominating set.

Dealing with the NP-complete Dominating Set problem on graphs, we demonstrate the power of data reduction by preprocessing from a theoretical as well as a practical side. In particular, we prove that Dominating Set restricted to planar graphs has a so-called problem kernel of linear size, achieved by two simple and easy to implement reduction rules. Moreover, having implemented our reduction rules, first experiments indicate the impressive practical potential of these rules. Thus, this work seems to open up a new and prospective way how to cope with one of the most important problems in graph theory and combinatorial optimization.

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Abstract. Determining whether a parameterized problem is kernelizable and has a small kernel size has recently become one of the most interesting topics of research in the area of parameterized complexity and algorithms. Theoretically, it has been proved that a parameterized problem is kernelizable if and only if it is fixed-parameter tractable. Practically, applying a data reduction algorithm to reduce an instance of a parameterized problem to an equivalent smaller instance (i.e., a kernel) has led to very efficient algorithms and now goes hand-in-hand with the design of practical algorithms for solving NP-hard problems. Well-known examples of such parameterized problems include the vertex cover problem, which is kernelizable to a kernel of size bounded by 2k, and the planar dominating set problem, which is kernelizable to a kernel of size bounded by 335k. In this paper we develop new techniques to derive upper and lower bounds on the kernel size for certain parameterized problems. In terms of our lower bound results, we show, for example, that unless P = NP, planar vertex cover does not have a problem kernel of size smaller than 4k/3, and planar independent set and planar dominating set do not have kernels of size smaller than 2k. In terms of our upper bound results, we further reduce the upper bound on the kernel size for the planar dominating set problem to 67k, improving significantly the 335k previous upper bound given by Alber, Fellows, and Niedermeier [J. ACM, 51 (2004), pp. 363–384]. This latter result is obtained by introducing a new set of reduction and coloring rules, which allows the derivation of nice combinatorial properties in the kernelized graph leading to a tighter bound on the size of the kernel. The paper also shows how this improved upper bound yields a simple and competitive algorithm for the planar dominating set problem.

We present an improved dynamic programming strategy for dominating set and related problems on graphs that are given together with a tree decomposition of width k. We obtain an O(4 n) algorithm for dominating set, where n is the number of nodes of the tree decomposition.

Abstract. A divide-and-conquer strategy based on variations of the Lipton-Tarjan planar separator theorem has been one of the most common approaches for solving planar graph problems for more than 20 years. We present a new framework for designing fast subexponential exact and parameterized algorithms on planar graphs. Our approach is based on geometric properties of planar branch decompositions obtained by Seymour & Thomas, combined with refined techniques of dynamic programming on planar graphs based on properties of non-crossing partitions. Compared to divide-and-conquer algorithms, the main advantages of our method are a) it is a generic method which allows to attack broad classes of problems; b) the obtained algorithms provide a better worst case analysis. To exemplify our approach we show how to obtain an O(26.903pn) time algorithm solving weighted Hamiltonian Cycle. We observe how our technique can be used to solve Planar Graph TSP in time O(29.8594pn). Our approach can be used to design parameterized algorithms as well. For example we introduce the first 2O(pk)nO(1) time algorithm for parameterized Planar k-cycle by showing that for a given k we can decide if a planar graph on n vertices has a cycle of length at least k in time O(213.6pkn + n3). 1 Introduction The celebrated Lipton-Tarjan planar separator theorem [20] is one of the most common approaches to obtain algorithms with subexponential running time for many problems on planar graphs [21]. The? This work was partially supported by Norges forskningsr*ad project 160778/V30,

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Dealing with the NP-complete Dominating Set problem on undirected graphs, we demonstrate the power of data reduction by preprocessing from a theoretical as well as a practical side. In particular, we prove that Dominating Set on planar graphs has a so-called problem kernel of linear size, achieved by two simple and easy to implement reduction rules. This answers an open question from previous work on the parameterized complexity of Dominating Set on planar graphs.

this paper was presented at the 11th Annual International Symposium on Algorithms And Computation (ISAAC'00), Springer-Verlag, LNCS 1969, pages 180--191, held in Taipei, Taiwan, December 2000. This conference version, however, contains a faulty application of the main result to the case of minimum weight vertex covers with a bound on the number of vertices