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Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees

Author

Listed:
  • Xinyue Liu

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

  • Huiqin Jiang

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

  • Pu Wu

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

  • Zehui Shao

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

Abstract

For a simple graph G = ( V , E ) with no isolated vertices, a total Roman {3}-dominating function(TR3DF) on G is a function f : V ( G ) → { 0 , 1 , 2 , 3 } having the property that (i) ∑ w ∈ N ( v ) f ( w ) ≥ 3 if f ( v ) = 0 ; (ii) ∑ w ∈ N ( v ) f ( w ) ≥ 2 if f ( v ) = 1 ; and (iii) every vertex v with f ( v ) ≠ 0 has a neighbor u with f ( u ) ≠ 0 for every vertex v ∈ V ( G ) . The weight of a TR3DF f is the sum f ( V ) = ∑ v ∈ V ( G ) f ( v ) and the minimum weight of a total Roman {3}-dominating function on G is called the total Roman {3}-domination number denoted by γ t { R 3 } ( G ) . In this paper, we show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Finally, we present a linear-time algorithm to compute the value of γ t { R 3 } for trees.

Suggested Citation

  • Xinyue Liu & Huiqin Jiang & Pu Wu & Zehui Shao, 2021. "Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees," Mathematics, MDPI, vol. 9(3), pages 1-7, February.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:3:p:293-:d:491394
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    References listed on IDEAS

    as
    1. H. Abdollahzadeh Ahangar & Michael A. Henning & Christian Löwenstein & Yancai Zhao & Vladimir Samodivkin, 2014. "Signed Roman domination in graphs," Journal of Combinatorial Optimization, Springer, vol. 27(2), pages 241-255, February.
    2. Wyatt J. Desormeaux & Teresa W. Haynes & Michael A. Henning, 2013. "Edge lifting and total domination in graphs," Journal of Combinatorial Optimization, Springer, vol. 25(1), pages 47-59, January.
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