engleski

Some Results for Roman Domination Numbers on Cardinal Products of Paths and Cycles

For a graph $G=(V, E)$, \emph{;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 \emph{;a}; $\gamma_R$\emph{;-function}; and its weight, denoted by $\gamma_R(G)$, is called \emph{;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 $\displaystyle{;\lim_{;m, n\to \infty};\frac{;\gamma_R(P_m\times P_n)};{;mn};};$ 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 $\displaystyle{;\lim_{;m, n\to \infty};\frac{;\gamma_R(C_m\times C_n)};{;mn};};$.

Roman dominating function, Roman domination number $\gamma_R$, cardinal product of paths, cardinal product of cycles

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