On perfect Roman domination number in trees: complexity and bounds

  • Mahsa Darkooti
  • Abdollah AlhevazEmail author
  • Sadegh Rahimi
  • Hadi Rahbani


A perfect Roman dominating function on a graph \(G =(V,E)\) is a function \(f: V \longrightarrow \{0, 1, 2\}\) satisfying the condition that every vertex u with \(f(u) = 0\) is adjacent to exactly one vertex v for which \(f(v)=2\). The weight of a perfect Roman dominating function f is the sum of the weights of the vertices. The perfect Roman domination number of G, denoted by \(\gamma _{R}^{p}(G)\), is the minimum weight of a perfect Roman dominating function in G. In this paper, we first show that the decision problem associated with \(\gamma _{R}^{p}(G)\) is NP-complete for bipartite graphs. Then, we prove that for every tree T of order \(n\ge 3\), with \(\ell \) leaves and s support vertices, \(\gamma _R^P(T)\le (4n-l+2s-2)/5\), improving a previous bound given in Henning et al. (Discrete Appl Math 236:235–245, 2018).


Dominating set Roman dominating function Perfect Roman dominating function 

Mathematics Subject Classification




The authors would like to thank both referees for their careful review of the paper and for valuable comments. Also, the authors would like to express their deep gratitude to Dr. Nader Jafari-Rad for a number of helpful suggestions. This research was in part supported by a grant from Shahrood University of Technology, Iran.


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Mahsa Darkooti
    • 1
  • Abdollah Alhevaz
    • 1
    Email author
  • Sadegh Rahimi
    • 1
  • Hadi Rahbani
    • 1
  1. 1.Faculty of Mathematical SciencesShahrood University of TechnologyShahroodIran

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