چكيده فارسي :
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
