On the total $(k,r)$-domination number of random graphs

A subset $S$ of a vertex set of a graph $G$ is a total $(k,r)$-dominating set if every vertex $u\in V(G)$ is within distance $k$ of at least $r$ vertices in $S$. The minimum cardinality among all total $(k,r)$-dominating sets of $G$ is called the total $(k,r)$-domination number of $G$, denoted by ${\gamma }_{(k,r)}^t(G)$. We previously gave an upper bound on ${\gamma }_{(2,r)}^t(G(n,p))$ in random graphs with non-fixed $p\in (0,1)$. In this paper we generalize this result to give an upper bound on ${\gamma }_{(k,r)}^t(G(n,p))$ in random graphs with non-fixed $p\in (0,1)$ for $k\ge 3$ as well as present an upper bound on ${\gamma }_{(k,r)}^t(G)$ in graphs with large girth.