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

Abstract

Given a positive integer k and a graph G = (V, E), a function f from V to the power set of Ik is called a k-rainbow function if for each vertex v ∈ V, f(v) = 0 implies ∪u∈N(v)f(u) = Ik where N(v) is the set of all neighbors of vertex v and Ik = {1,…, k}. Finding a k-rainbow function of minimum weight of Σv∈V |f(v)|, 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 tree-width tw in O ((2k+1 + 1)tw w 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. © 2021