Independent dominating set problem revisited

An independent dominating set of a graph G is a subset D of V such that every vertex not in D is adjacent to at least one vertex of D and no two vertices in D are adjacent. The independent dominating set (IDS) problem   asks for an independent dominating set with minimum cardinality. First, we show that the independent dominating set problem and the dominating set problem on cubic bipartite graphs are both NP-complete. As an additional result, we give an alternative and more direct proof for the NP-completeness of both the independent dominating set problem and the dominating set problem on at-most-cubic grid graphs. Next, we show that there are fixed-parameter tractable algorithms for the independent dominating set problem and the dominating set problem on at-most-cubic graphs, which run in O(3.3028k+n)O(3.3028k+n) and O(4.2361k+n)O(4.2361k+n) time, respectively. Moreover, we consider the weighted independent dominating set problem on (k,ℓ)(k,ℓ)-graphs. We show that the problem on (2,1)(2,1)-graphs is NP-complete. We also show that the problem can be solved in linear time for (1,1)(1,1)-graphs and in polynomial time for (1,ℓ)(1,ℓ)-graphs for constant ℓ, respectively.