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Independent Rainbow Domination of Graphs

  • Zehui Shao
  • Zepeng Li
  • Aljoša Peperko
  • Jiafu Wan
  • Janez Žerovnik
Article

Abstract

Given a positive integer t and a graph F, the goal is to assign a subset of the color set \(\{1,2,\ldots ,t\}\) to every vertex of F such that every vertex with the empty set assigned has all t colors in its neighborhood. Such an assignment is called the t-rainbow dominating function (\(t\mathrm{RDF}\)) of the graph F. A \(t\mathrm{RDF}\) is independent (\(It\mathrm{RDF}\)) if vertices assigned with non-empty sets are pairwise non-adjacent. The weight of a \(t\mathrm{RDF}\) g of a graph F is the value \(w(g) =\sum _{v \in V(F)}|g(v)|\). The independent t-rainbow domination number \(i_{rt}(F)\) is the minimum weight over all \(It\mathrm{RDF}\)s of F. In this article, it is proved that the independent t-rainbow domination problem is NP-complete even if the input graph is restricted to a bipartite graph or a planar graph, and the results of the study provide some bounds for the independent t-rainbow domination number of any graph for a positive integer t. Moreover, the exact values and bounds of the independent t-rainbow domination numbers of some Petersen graphs and torus graphs are given.

Keywords

Rainbow domination Domination number Independent rainbow domination NP-complete 

Mathematics Subject Classification

05C69 05C15 05C76 05C85 

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Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2017

Authors and Affiliations

  1. 1.School of Computer Science and Educational SoftwareGuangzhou UniversityGuangzhouChina
  2. 2.School of Information Science and EngineeringChengdu UniversityChengduChina
  3. 3.School of Electronic Engineering and Computer SciencePeking UniversityBeijingChina
  4. 4.FME, University of LjubljanaLjubljanaSlovenia
  5. 5.Institute of Mathematics, Physics and MechanicsLjubljanaSlovenia
  6. 6.School of Mechanical and Automotive EngineeringSouth China University of TechnologyGuangzhouChina

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