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 I k is called a k -rainbow function if for each vertex v ∈ V , f ( v )=∅ implies ∪ u ∈ 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 ∑ v ∈ 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 t w in 𝒪((2 k + 1 +1) t 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.