Electronic Journal of Graph Theory and Applications (Oct 2021)

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

  • M. Alambardar Meybodi,
  • M. R. Hooshmandasl,
  • P. Sharifani,
  • Ali Shakiba

DOI
https://doi.org/10.5614/ejgta.2021.9.2.4
Journal volume & issue
Vol. 9, no. 2
pp. 277 – 300

Abstract

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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)=∅ 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 ∑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 tw in 𝒪((2k + 1+1)twn) 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.

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