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Algorithmic results on double Roman domination in graphs

Author

Listed:
  • S. Banerjee

    (Indian Institute of Technology (ISM))

  • Michael A. Henning

    (University of Johannesburg)

  • D. Pradhan

    (Indian Institute of Technology (ISM))

Abstract

Given a graph $$G=(V,E)$$G=(V,E), a function $$f:V\longrightarrow \{0,1,2,3\}$$f:V⟶{0,1,2,3} is called a double Roman dominating function on G if (i) for every $$v\in V$$v∈V with $$f(v)=0$$f(v)=0, there are at least two neighbors of v that are assigned 2 under f or at least a neighbor of v that is assigned 3 under f, and (ii) for every vertex v with $$f(v)=1$$f(v)=1, there is at least one neighbor w of v with $$f(w)\ge 2$$f(w)≥2. The weight of a double Roman dominating function f is $$f(V)=\sum _{u\in V}f(u)$$f(V)=∑u∈Vf(u). The double Roman domination number of G, denoted by $$\gamma _{dR}(G)$$γdR(G) is the minimum weight of a double Roman dominating function on G. For a graph $$G=(V,E)$$G=(V,E), Min-Double-RDF is to find a double Roman dominating function f with $$f(V)=\gamma _{dR}(G)$$f(V)=γdR(G). The decision version of Min-Double-RDF is shown to be NP-complete for chordal graphs and bipartite graphs. In this paper, we first strengthen the known NP-completeness of the decision version of Min-Double-RDF by showing that the decision version of Min-Double-RDF remains NP-complete for undirected path graphs, chordal bipartite graphs, and circle graphs. We then present linear time algorithms for computing the double Roman domination number in proper interval graphs and block graphs. We then discuss on the approximability of Min-Double-RDF and present a 2-approximation algorithm in 3-regular bipartite graphs.

Suggested Citation

  • S. Banerjee & Michael A. Henning & D. Pradhan, 2020. "Algorithmic results on double Roman domination in graphs," Journal of Combinatorial Optimization, Springer, vol. 39(1), pages 90-114, January.
  • Handle: RePEc:spr:jcomop:v:39:y:2020:i:1:d:10.1007_s10878-019-00457-3
    DOI: 10.1007/s10878-019-00457-3
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    References listed on IDEAS

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    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. Yue, Jun & Wei, Meiqin & Li, Min & Liu, Guodong, 2018. "On the double Roman domination of graphs," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 669-675.
    3. J. Amjadi & S. Nazari-Moghaddam & S. M. Sheikholeslami & L. Volkmann, 2018. "An upper bound on the double Roman domination number," Journal of Combinatorial Optimization, Springer, vol. 36(1), pages 81-89, July.
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    Cited by:

    1. Peng Li, 2024. "The k-th Roman domination problem is polynomial on interval graphs," Journal of Combinatorial Optimization, Springer, vol. 48(3), pages 1-14, October.
    2. Darja Rupnik Poklukar & Janez Žerovnik, 2023. "Double Roman Domination: A Survey," Mathematics, MDPI, vol. 11(2), pages 1-20, January.
    3. Ching-Chi Lin & Cheng-Yu Hsieh & Ta-Yu Mu, 2022. "A linear-time algorithm for weighted paired-domination on block graphs," Journal of Combinatorial Optimization, Springer, vol. 44(1), pages 269-286, August.
    4. Zehui Shao & Rija Erveš & Huiqin Jiang & Aljoša Peperko & Pu Wu & Janez Žerovnik, 2021. "Double Roman Graphs in P (3 k , k )," Mathematics, MDPI, vol. 9(4), pages 1-18, February.
    5. Ana Klobučar Barišić & Robert Manger, 2024. "Solving the minimum-cost double Roman domination problem," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 32(3), pages 793-817, September.

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