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(c ₁ ^ \sqrt k n) time, where c ₁ =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.