Refined Search Tree Technique for DOMINATING SET on Planar Graphs

We establish a refined search tree technique for the parameterized Dominating Set problem on planar graphs. Here, we are given an undirected graph and we ask for a set of at most k vertices such that each other graph vertex has at least one neighbor in this set. We derive search-tree based fixed-parameter solving algorithms with running times O(8n) and O(8k + n), where n is the number of vertices in the graph. For our search tree, we firstly provide a set of reduction rules. Secondly, we prove an intricate branching theorem based on the Euler formula. In addition, we give a family of example graphs showing that the bound of the branching theorem is optimal with respect to our reduction rules. Our final search tree is very easy (to implement); its correctness analysis, however, is involved.