Doost Ali Mojdeh

Let $G$ be a graph with vertex set $V=V(G)$ and edge set $E=E(G)$. A subset $S$ of $V$ is a total $k$-dominating set of $G$ if every vertex in $V$ is within distance $k$ from some vertex in $S$. The total $k$-domination number, $\gamma_{t}^{k}(G)$ is the minimum cardinality of a total $k$-dominating set of $G$. We are interested in to introduce two types of total $k$-dominating sets, namely locating-total $k$-dominating set of a graph $G$ ($\gamma_{t,k}^{L}(G)$-set) and differentiating-total $k$-dominating set of a graph $G$ ($\gamma_{t,k}^{D}(G)$-set). We show that; for any graph $G$ of order $n$, $\log_2 {n}-1\leq \gamma_{t,k}^{L}(G)\leq \frac{3}{5}n$. We obtain sharp upper and lower bounds for $\gamma_{t,k}^{L}(T)$ and $\gamma_{t,k}^{D}(T)$ for every tree $T$.