On the k-rainbow domination in graphs with bounded tree-width

Given a positive integer k and a graph G = (V, E), a function f from V to the power set of I-k is called a k-rainbow function if for each vertex v is an element of V, f(v) = empty set implies boolean OR(u is an element of N(v))f(u) = I-k where N(v) is the set of all neighbors of vertex v and I-k = {1, ..., k}. Finding a k-rainbow function of minimum weight of Sigma(v is an element of V) vertical bar f(v vertical bar|, which is called the k-rainbow domination problem, is known to be NP-complete for arbitrary graphs and values of k. In this paper, we propose a dynamic programming algorithm to solve the k-rainbow domination problem for graphs with bounded treewidth tw in O((2(k+1) + 1)(tw) n) time, where G has n vertices. Moreover, we also show that the same approach is applicable to solve the weighted k-rainbow domination problem with the same complexity. Therefore, both problems of k-rainbow and weighted k-rainbow domination belong to the class FPT, or fixed parameter tractable, with respect to tree-width. In addition to formally showing the correctness of our algorithms, we also implemented these algorithms to illustrate some examples.