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For any graph $G$ of order $n$ with degree sequence $d_{1}\geq\cdots\geqd_{n}$, we define the double Slater number $s\ell_{\times2}(G)$ as the smallestinteger $t$ such that $t+d_{1}+\cdots+d_{t-e}\geq2n-p$ in which $e$ and $p$ arethe number of end-vertices and penultimate vertices of $G$, respectively. Weshow 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 theequality 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 improvingand generalizing some earlier results on double domination in graphs. We givean upper bound on the $k$-tuple domatic number of graphs with characterizationof all graphs attaining the bound. Finally, we characterize the family of allfull graphs, leading to a solution to an open problem given in a paper byCockayne and Hedetniemi ($1977$).

Revisiting $k$-tuple dominating sets with emphasis on small values of $k$