Henning Fernau

Fixed Parameter Algorithms for DOMINATING SET and Related Problems on Planar Graphs

Abstract. We present an algorithm that constructively produces a solution to the k -DOMINATING SET problem for planar graphs in time O(c^ \sqrt k n) , where c=4^ 6\sqrt 34 . To obtain this result, we show that the treewidth of a planar graph with domination number γ (G) is O(\sqrt \rule 0pt 4pt \smash γ (G) ) , and that such a tree decomposition can be found in O(\sqrt \rule 0pt 4pt \smash γ (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(c1^ \sqrt k n) time, where c1=3^ 36\sqrt 34 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.

Parametric Duality and Kernelization: Lower Bounds and Upper Bounds on Kernel Size

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

Kernel(s) for problems with no kernel: On out-trees with many leaves

\mathcal{NP}

Fixed Parameter Algorithms for PLANAR DOMINATING SET and Related Problems

-hard problems. Well-known examples of such parameterized problems include the vertex cover problem, which is kernelizable to a kernel of size bounded by

EDGE DOMINATING SET: efficient enumeration-based exact algorithms

2k

Graph separators: a parameterized view

, and the planar dominating set problem, which is kernelizable to a kernel of size bounded by

A Top-Down Approach to Search-Trees: Improved Algorithmics for 3-Hitting Set

335k

Vertex and edge covers with clustering properties: Complexity and algorithms

. 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

On Parameterized Enumeration

\mathcal{P} = \mathcal{NP}

Refined Search Tree Technique for DOMINATING SET on Planar Graphs

, planar vertex cover does not have a problem kernel of size smaller than