# Strong Roman Domination in Grid Graphs

• Chen, Xue-Gang (Department of Mathematics, North China Electric Power University) ;
• Sohn, Moo Young (Department of Mathematics, Changwon National University)
• Accepted : 2019.09.19
• Published : 2019.09.23
• 13 2

#### Abstract

Consider a graph G of order n and maximum degree ${\Delta}$. Let $f:V(G){\rightarrow}\{0,1,{\cdots},{\lceil}{\frac{{\Delta}}{2}}{\rceil}+1\}$ be a function that labels the vertices of G. Let $B_0=\{v{\in}V(G):f(v)=0\}$. The function f is a strong Roman dominating function for G if every $v{\in}B_0$ has a neighbor w such that $f(w){\geq}1+{\lceil}{\frac{1}{2}}{\mid}N(w){\cap}B_0{\mid}{\rceil}$. In this paper, we study the bounds on strong Roman domination numbers of the Cartesian product $P_m{\square}P_k$ of paths $P_m$ and paths $P_k$. We compute the exact values for the strong Roman domination number of the Cartesian product $P_2{\square}P_k$ and $P_3{\square}P_k$. We also show that the strong Roman domination number of the Cartesian product $P_4{\square}P_k$ is between ${\lceil}{\frac{1}{3}}(8k-{\lfloor}{\frac{k}{8}}{\rfloor}+1){\rceil}$ and ${\lceil}{\frac{8k}{3}}{\rceil}$ for $k{\geq}8$, and that both bounds are sharp bounds.

#### Keywords

Roman domination number;strong Roman domination number;grid

#### Acknowledgement

Supported by : Changwon National University

#### References

1. M P. Alvarez-Ruiz, T. Mediavilla-Gradolph, S. M. Sheikholeslami, J. C. Valenzuela-Tripodoro and I. G. Yero, On the strong Roman domination number of graphs, Discrete Appl. Math., 231(2017), 44-59. https://doi.org/10.1016/j.dam.2016.12.013
2. E. J. Cockayne, P. M. Dreyer Jr., S. M. Hedetniemi and S. T. Hedetniemi, Roman domination in graphs, Discrete Math., 278(2004), 11-22. https://doi.org/10.1016/j.disc.2003.06.004
3. C. S. ReVelle and K. E. Rosing, Defendents imperium romanum: a classical problem in military strategy, Amer. Math. Monthly, 107(7)(2000), 585-594. https://doi.org/10.1080/00029890.2000.12005243
4. I. Stewart, Defend the roman empire!, Sci. Amer., 281(6)(1999), 136-138. https://doi.org/10.1038/scientificamerican1299-136