We present an algorithm that constructively produces a solution to the k-DOMINATING SET problem for planar graphs in time O(c √ k n), where c = 4 6 √ 34 . To obtain this result, we show that the treewidth of a planar graph with domination number γ (G) is O( √ γ (G)), and that such a tree decomposition can be found in O( √ 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(c √ k 1 n) time, where c 1 = 3 36 √ 34and 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.

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