  In this paper, we study the most basic domination invariants in graphs, in
which number 2 is intrinsic part of their definitions. We classify them upon
three criteria, two of which give the following previously studied invariants:
the weak $2$-domination number, $\gamma_{w2}(G)$, the $2$-domination number,
$\gamma_2(G)$, the $\{2\}$-domination number, $\gamma_{\{2\}}(G)$, the double
domination number, $\gamma_{\times 2}(G)$, the total $\{2\}$-domination number,
$\gamma_{t\{2\}}(G)$, and the total double domination number, $\gamma_{t\times
2}(G)$, where $G$ is a graph in which a corresponding invariant is well
defined. The third criterion yields rainbow versions of the mentioned six
parameters, one of which has already been well studied, and three other give
new interesting parameters. Together with a special, extensively studied Roman
domination, $\gamma_R(G)$, and two classical parameters, the domination number,
$\gamma(G)$, and the total domination number, $\gamma_t(G)$, we consider 13
domination invariants in graphs $G$. In the main result of the paper we present
sharp upper and lower bounds of each of the invariants in terms of every other
invariant, large majority of which are new results proven in this paper. As a
consequence of the main theorem we obtain some complexity results for the
studied invariants, in particular regarding the existence of approximation
algorithms and inapproximability bounds.
