Fixed Parameter Algorithms for PLANAR DOMINATING SET and Related Problems

We present an algorithm that constructively produces a solution to the k-dominating set problem for planar graphs in time O(c √ n), where c = 36 √ 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 + n 2) time, where c1 = 2 36 √ 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.