Lower bounds for the total (distance) $k$-domination number of a graph

For $k \geq 1$ and a graph $G$ without isolated vertices, a \emph{total (distance) $k$-dominating set} of $G$ is a set of vertices $S \subseteq V(G)$ such that every vertex in $G$ is within distance $k$ to some vertex of $S$ other than itself. The \emph{total (distance) $k$-domination number} of $G$ is the minimum cardinality of a total $k$-dominating set in $G$, and is denoted by $\gamma{k}^t(G)$. When $k=1$, the total $k$-domination number reduces to the \emph{total domination number}, written $\gammat(G)$; that is, $\gammat(G) = \gamma{1}^t(G)$. This paper shows that several known lower bounds on the total domination number generalize nicely to lower bounds on total (distance) $k$-domination.