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Total Roman 2-Reinforcement of Graphs

Author

Listed:
  • M. Kheibari
  • H. Abdollahzadeh Ahangar
  • R. Khoeilar
  • S. M. Sheikholeslami
  • Ahmet Sinan Cevik

Abstract

A total Roman 2-dominating function (TR2DF) on a graph Γ=V,E is a function l:V⟶0,1,2, satisfying the conditions that (i) for every vertex y∈V with ly=0, either y is adjacent to a vertex labeled 2 under l, or y is adjacent to at least two vertices labeled 1; (ii) the subgraph induced by the set of vertices with positive weight has no isolated vertex. The weight of a TR2DF l is the value ∑y∈Vly. The total Roman 2-domination number (TR2D-number) of a graph Γ is the minimum weight of a TR2DF on Γ. The total Roman 2-reinforcement number (TR2R-number) of a graph is the minimum number of edges that have to be added to the graph in order to decrease the TR2D-number. In this manuscript, we study the properties of TR2R-number and we present some sharp upper bounds. In particular, we determine the exact value of TR2R-numbers of some classes of graphs.

Suggested Citation

  • M. Kheibari & H. Abdollahzadeh Ahangar & R. Khoeilar & S. M. Sheikholeslami & Ahmet Sinan Cevik, 2021. "Total Roman 2-Reinforcement of Graphs," Journal of Mathematics, Hindawi, vol. 2021, pages 1-7, April.
    • Handle: RePEc:hin:jjmath:5515250
    DOI: 10.1155/2021/5515250
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