  For an integer $k \ge 1$, a (distance) $k$-dominating set of a connected
graph $G$ is a set $S$ of vertices of $G$ such that every vertex of $V(G)
\setminus S$ is at distance at most~$k$ from some vertex of $S$. The
$k$-domination number, $\gamma_k(G)$, of $G$ is the minimum cardinality of a
$k$-dominating set of $G$. In this paper, we establish lower bounds on the
$k$-domination number of a graph in terms of its diameter, radius and girth. We
prove that for connected graphs $G$ and $H$, $\gamma_k(G \times H) \ge
\gamma_k(G) + \gamma_k(H) -1$, where $G \times H$ denotes the direct product of
$G$ and $H$.
