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On the double Roman domination of graphs

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  • Yue, Jun
  • Wei, Meiqin
  • Li, Min
  • Liu, Guodong

Abstract

A double Roman dominating function of a graph G is a labeling f: V(G) → {0, 1, 2, 3} such that if f(v)=0, then the vertex v must have at least two neighbors labeled 2 under f or one neighbor with f(w)=3, and if f(v)=1, then v must have at least one neighbor with f(w) ≥ 2. The double Roman domination number γdR(G) of G is the minimum value of Σv ∈ V(G)f(v) over such functions. In this paper, we firstly give some bounds of the double Roman domination numbers of graphs with given minimum degree and graphs of diameter 2, and further we get that the double Roman domination numbers of almost all graphs are at most n. Then we obtain sharp upper and lower bounds for γdR(G)+γdR(G¯). Moreover, a linear time algorithm for the double Roman domination number of a cograph is given and a characterization of the double Roman cographs is provided. Those results partially answer two open problems posed by Beeler et al. (2016).

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:338:y:2018:i:c:p:669-675
    DOI: 10.1016/j.amc.2018.06.033
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    References listed on IDEAS

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    1. Yue, Jun, 2016. "Acyclic and star coloring of P4-reducible and P4-sparse graphs," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 68-73.
    2. Hengzhe Li & Yuxing Yang & Baoyindureng Wu, 2016. "2-Edge connected dominating sets and 2-Connected dominating sets of a graph," Journal of Combinatorial Optimization, Springer, vol. 31(2), pages 713-724, February.
    3. Yongtang Shi & Meiqin Wei & Jun Yue & Yan Zhao, 2017. "Coupon coloring of some special graphs," Journal of Combinatorial Optimization, Springer, vol. 33(1), pages 156-164, January.
    4. Chen, He & Jin, Zemin, 2017. "Coupon coloring of cographs," Applied Mathematics and Computation, Elsevier, vol. 308(C), pages 90-95.
    5. Chen, Lily & Ma, Yingbin & Shi, Yongtang & Zhao, Yan, 2018. "On the [1,2]-domination number of generalized Petersen graphs," Applied Mathematics and Computation, Elsevier, vol. 327(C), pages 1-7.
    6. Yue, Jun & Zhang, Shiliang & Zhang, Xia, 2016. "Note on the perfect EIC-graphs," Applied Mathematics and Computation, Elsevier, vol. 289(C), pages 481-485.
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    Cited by:

    1. Frank Werner, 2019. "Discrete Optimization: Theory, Algorithms, and Applications," Mathematics, MDPI, vol. 7(5), pages 1-4, May.
    2. Enrico Enriquez & Grace Estrada & Carmelita Loquias & Reuella J Bacalso & Lanndon Ocampo, 2021. "Domination in Fuzzy Directed Graphs," Mathematics, MDPI, vol. 9(17), pages 1-14, September.
    3. Samadi, B. & Soltankhah, N. & Abdollahzadeh Ahangar, H. & Chellali, M. & Mojdeh, D.A. & Sheikholeslami, S.M. & Valenzuela-Tripodoro, J.C., 2023. "Restrained condition on double Roman dominating functions," Applied Mathematics and Computation, Elsevier, vol. 438(C).
    4. Liu, Xiaoxiao & Sun, Shiwen & Wang, Jiawei & Xia, Chengyi, 2019. "Onion structure optimizes attack robustness of interdependent networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 535(C).
    5. Darja Rupnik Poklukar & Janez Žerovnik, 2023. "Double Roman Domination: A Survey," Mathematics, MDPI, vol. 11(2), pages 1-20, January.
    6. 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.
    7. Bermudo, Sergio & Higuita, Robinson A. & Rada, Juan, 2020. "Domination in hexagonal chains," Applied Mathematics and Computation, Elsevier, vol. 369(C).
    8. Hao Guan & Waheed Ahmad Khan & Amna Fida & Khadija Ali & Jana Shafi & Aysha Khan, 2024. "Dominations in Intutionistic Fuzzy Directed Graphs with Applications towards Influential Graphs," Mathematics, MDPI, vol. 12(6), pages 1-14, March.
    9. Banerjee, S. & Henning, Michael A. & Pradhan, D., 2021. "Perfect Italian domination in cographs," Applied Mathematics and Computation, Elsevier, vol. 391(C).
    10. Ma, Yuede & Cai, Qingqiong & Yao, Shunyu, 2019. "Integer linear programming models for the weighted total domination problem," Applied Mathematics and Computation, Elsevier, vol. 358(C), pages 146-150.
    11. 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.

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