Bounds on the Distance Roman domination Number in graphs

‎For a graph $G$ and positive integers $k$ and $r$‎, ‎a function $f:V(G)\rightarrow \{0,1,2\}$ is a‎ ‎\textit{distance-$k$ Roman $r$-dominating function} if every‎ ‎vertex $u$ for which $f(u)=0$ is within distance $k$ of at least‎ ‎$r$ vertices $v$ for which $f(v)=2$‎. ‎The weight of a distance-$k$‎ ‎Roman $r$-dominating function is the sum of labels attributed to all vertices‎. ‎The \textit{distance-$k$ Roman $r$-domination number}‎ ‎of a graph $G$‎, ‎denoted by $\gamma_{R}^{(k,r)}(G)$‎, ‎is the‎ ‎minimum weight of a distance-$k$ Roman $r$-dominating function on‎ ‎$G$‎. ‎We present probabilistic bounds for the distance-$k$ Roman $r$-domination number of a graph $G$‎, ‎and then we study this parameter in random graphs‎