  For any graph $G$ of order $n$ with degree sequence $d_{1}\geq\cdots\geq
d_{n}$, we define the double Slater number $s\ell_{\times2}(G)$ as the smallest
integer $t$ such that $t+d_{1}+\cdots+d_{t-e}\geq2n-p$ in which $e$ and $p$ are
the number of end-vertices and penultimate vertices of $G$, respectively. We
show that $\gamma_{\times2}(G)\geq s\ell_{\times2}(G)$, where
$\gamma_{\times2}(G)$ is the well-known double domination number of a graph $G$
with no isolated vertices. We prove that the problem of deciding whether the
equality holds for a given graph is NP-complete even when restricted to
$4$-partite graphs. We also prove that the problem of computing
$\gamma_{\times2}(G)$ in NP-hard even for comparability graphs of diameter two.
Some results concerning these two parameters are given in this paper improving
and generalizing some earlier results on double domination in graphs. We give
an upper bound on the $k$-tuple domatic number of graphs with characterization
of all graphs attaining the bound. Finally, we characterize the family of all
full graphs, leading to a solution to an open problem given in a paper by
Cockayne and Hedetniemi ($1977$).
