Geodesic
Some Results for Roman Domination Number on Cardinal Product of Paths and Cycles
Kragujevac Journal of Mathematics, Tome 38 (2014) no. 1, p. 83

Voir la notice de l'article provenant de la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

For a graph $G=(V,E)$, a Roman dominating function (RDF) is a function $f \colon V \to \{0,1,2\}$ satisfying the condition that every vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=2$. The weight of an RDF equals $w(f)=\sum_{v\in V}f(v)=|V_1|+2|V_2|$ where $V_i=\{v\in V: f(v)=i\}$, $i\in \{1,2\}$. An RDF for which $w(f)$ achieves its minimum is called a $\gamma_R$-function and its weight, denoted by $\gamma_R(G)$, is called the Roman domination number. In this paper we determine a lower and the upper bounds for $\gamma_R(P_m\times P_n)$ as well as the exact value of $\lim_{m,n\to \infty} \frac{\gamma_R(P_m\times P_n)}{mn}$ where $P_m\times P_n$ stands for the cardinal product of two paths. We also present some results concerning the cardinal product of two cycles $C_m\times C_n$ as well as the exact value of $\lim_{m,n\to \infty}\frac{\gamma_R(C_m\times C_n)}{mn}$.
Classification : 05C69 05C38
Keywords: Roman dominating function, Roman domination number $\gamma_R$, Cardinal product of paths, Cardinal product of cycles 
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     author = {A. Klobu\v{c}ar and I. Pulji\'c},
     title = {Some {Results} for {Roman} {Domination} {Number} on {Cardinal} {Product} of {Paths} and {Cycles}},
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A. Klobučar; I. Puljić. Some Results for Roman Domination Number on Cardinal Product of Paths and Cycles. Kragujevac Journal of Mathematics, Tome 38 (2014) no. 1, p. 83 . http://geodesic.mathdoc.fr/item/KJM_2014_38_1_a5/