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Signed Roman domination in graphs

Author

Listed:
  • H. Abdollahzadeh Ahangar

    (Babol University of Technology)

  • Michael A. Henning

    (University of Johannesburg)

  • Christian Löwenstein

    (University of Johannesburg)

  • Yancai Zhao

    (Wuxi City College of Vocational Technology)

  • Vladimir Samodivkin

    (University of Architecture Civil Engineering and Geodesy)

Abstract

In this paper we continue the study of Roman dominating functions in graphs. A signed Roman dominating function (SRDF) on a graph G=(V,E) is a function f:V→{−1,1,2} satisfying the conditions that (i) the sum of its function values over any closed neighborhood is at least one and (ii) for every vertex u for which f(u)=−1 is adjacent to at least one vertex v for which f(v)=2. The weight of a SRDF is the sum of its function values over all vertices. The signed Roman domination number of G is the minimum weight of a SRDF in G. We present various lower and upper bounds on the signed Roman domination number of a graph. Let G be a graph of order n and size m with no isolated vertex. We show that $\gamma _{\mathrm{sR}}(G) \ge\frac{3}{\sqrt{2}} \sqrt{n} - n$ and that γ sR(G)≥(3n−4m)/2. In both cases, we characterize the graphs achieving equality in these bounds. If G is a bipartite graph of order n, then we show that $\gamma_{\mathrm{sR}}(G) \ge3\sqrt{n+1} - n - 3$ , and we characterize the extremal graphs.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jcomop:v:27:y:2014:i:2:d:10.1007_s10878-012-9500-0
    DOI: 10.1007/s10878-012-9500-0
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    Citations

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    Cited by:

    1. Banerjee, S. & Henning, Michael A. & Pradhan, D., 2021. "Perfect Italian domination in cographs," Applied Mathematics and Computation, Elsevier, vol. 391(C).
    2. Hong Gao & Xing Liu & Yuanyuan Guo & Yuansheng Yang, 2022. "On Two Outer Independent Roman Domination Related Parameters in Torus Graphs," Mathematics, MDPI, vol. 10(18), pages 1-15, September.
    3. Lutz Volkmann, 2016. "Signed total Roman domination in graphs," Journal of Combinatorial Optimization, Springer, vol. 32(3), pages 855-871, October.
    4. Jia-Xiong Dan & Zhi-Bo Zhu & Xin-Kui Yang & Ru-Yi Li & Wei-Jie Zhao & Xiang-Jun Li, 2022. "The signed edge-domatic number of nearly cubic graphs," Journal of Combinatorial Optimization, Springer, vol. 44(1), pages 435-445, August.
    5. Guoliang Hao & Jianguo Qian, 2018. "Bounds on the domination number of a digraph," Journal of Combinatorial Optimization, Springer, vol. 35(1), pages 64-74, 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. 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.
    8. Mahsa Darkooti & Abdollah Alhevaz & Sadegh Rahimi & Hadi Rahbani, 2019. "On perfect Roman domination number in trees: complexity and bounds," Journal of Combinatorial Optimization, Springer, vol. 38(3), pages 712-720, October.
    9. H. Abdollahzadeh Ahangar & J. Amjadi & S. M. Sheikholeslami & L. Volkmann & Y. Zhao, 2016. "Signed Roman edge domination numbers in graphs," Journal of Combinatorial Optimization, Springer, vol. 31(1), pages 333-346, January.

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