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Total 2-Rainbow Domination in Graphs

Author

Listed:
  • Huiqin Jiang

    (Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China)

  • Yongsheng Rao

    (Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China)

Abstract

A total k-rainbow dominating function on a graph G = ( V , E ) is a function f : V ( G ) → 2 { 1 , 2 , … , k } such that (i) ∪ u ∈ N ( v ) f ( u ) = { 1 , 2 , … , k } for every vertex v with f ( v ) = ∅ , (ii) ∪ u ∈ N ( v ) f ( u ) ≠ ∅ for f ( v ) ≠ ∅ . The weight of a total 2-rainbow dominating function is denoted by ω ( f ) = ∑ v ∈ V ( G ) | f ( v ) | . The total k-rainbow domination number of G is the minimum weight of a total k-rainbow dominating function of G . The minimum total 2-rainbow domination problem (MT2RDP) is to find the total 2-rainbow domination number of the input graph. In this paper, we study the total 2-rainbow domination number of graphs. We prove that the MT2RDP is NP-complete for planar bipartite graphs, chordal bipartite graphs, undirected path graphs and split graphs. Then, a linear-time algorithm is proposed for computing the total k-rainbow domination number of trees. Finally, we study the difference in complexity between MT2RDP and the minimum 2-rainbow domination problem.

Suggested Citation

  • Huiqin Jiang & Yongsheng Rao, 2022. "Total 2-Rainbow Domination in Graphs," Mathematics, MDPI, vol. 10(12), pages 1-13, June.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:12:p:2059-:d:838600
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