Fixed Parameter Algorithms for Planar Dominating Set and Related Problems

Abstract

We present an algorithm for computing the domination number of a planar graph that uses $O\left( {c^{\sqrt k } n} \right)$ time, where k is the domination number of the given planar input graph and $c = 3^{6\sqrt {34} }$ . To obtain this result, we show that the treewidth of a planar graph with domination number k is $O\left( {\sqrt k } \right)$ , and that such a tree decomposition can be found in $O\left( {\sqrt {kn} } \right)$ time. The same technique can be used to show that the disk dimension problem (find a minimum set of faces that cover all vertices of a given plane graph) can be solved in $O\left( {c_1^{\sqrt {k_n } } } \right)$ time for $c_1 = 2^{6\sqrt {34} }$ . Similar results can be obtained for some variants of {updominating set}, e.g., INDEPENDENT DOMINATING SET.

Work supported by the DFG-research project PEAL (Parameterized complexity and Exact ALgorithms), NI 369/1-1.