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The Perfect Roman Domination Number of the Cartesian Product of Some Graphs

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  • Ahlam Almulhim
  • Abolape Deborah Akwu
  • Bana Al Subaiei
  • Akbar Ali

Abstract

A perfect Roman dominating function on a graph G is a function f:VG⟶0,1,2 for which every vertex v with fv=0 is adjacent to exactly one neighbor u with fu=2. The weight of f is the sum of the weights of the vertices. The perfect Roman domination number of a graph G, denoted by γRpG, is the minimum weight of a perfect Roman dominating function on G. In this paper, we prove that if G is the Cartesian product of a path Pr and a path Ps, a path Pr and a cycle Cs, or a cycle Cr and a cycle Cs, where r,s>5, then γRpG≤2/3G.

Suggested Citation

  • Ahlam Almulhim & Abolape Deborah Akwu & Bana Al Subaiei & Akbar Ali, 2022. "The Perfect Roman Domination Number of the Cartesian Product of Some Graphs," Journal of Mathematics, Hindawi, vol. 2022, pages 1-6, October.
    • Handle: RePEc:hin:jjmath:1957027
    DOI: 10.1155/2022/1957027
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