Given a graph $G$, a set of vertices $S\subseteq V(G)$ is a (resp. total or double) $2$-outer independent dominating set, if $S$ is a (resp. total or double) $2$-dominating set whose complement is an independent set. The (resp. total or double) $2$-outer independent domination number of $G$ is the smallest possible cardinality of a (resp. total or double) $2$-outer independent dominating set of $G$. In this paper, the $2$-outer independent, the total outer independent and the double outer independent domination numbers of graphs are investigated. We make some comparisons among these three domination parameters and bound their values from above and below. Moreover, we prove some Nordhaus-Gaddum type inequalities for them and present some complexity issues concerning finding their values.
