Let us temporarily say that a subset $A$ of (the set of vertices of) an undirected graph $G$ is dense when each vertex of $G$ is adjacent to an element of $A$. In other words, the (distance-$1$, open) neighborhood of $A$ is all of $G$.
Question 0: I suppose there is a more standard term for this than "dense", which I just made up. What is it?
Now let us consider the following invariant $\psi(G)$ of an undirected graph $G$: it is the largest number $n$ of pairwise disjoint dense subsets of $G$ (where "dense" is defined as above). I.e., it is the largest $n$ such that we can find $A_1,\ldots,A_n$ pairwise disjoint subsets of $G$ such that every vertex of $G$ is a adjacent to an element of each one of the $A_i$. (We agree that $\psi(G)=0$ if no dense subset exists, i.e., whenever some vertex is isolated.)
Evidently, $\psi(G)$ is at most the minimal degree $\delta(G)$ of $G$.
Question 1: Does $\psi(G)$ have a standard name and/or notation? If so, what is it?
Question 2: Is there some literature pertaining to this quantity?
Motivation: Consider the information theory protocol where Alice is given a vertex $x$ of $G$ and must choose a neighbor $x'$ of $x$ which she will communicate to her accomplice Bob (Bob is not given the value of $x$); Alice is trying to communicate some information to Bob: the obvious protocol (namely, Bob receives the information $i$ where $x'\in A_i$) will allow Alice to pass $\log n$ bits of information to Bob where $n = \psi(G)$. See here for a motivation of the motivation.
Edit: As Michal Adamaszek answers in the comments, the answer to question 0 is "total dominating" (in contrast, a "dominating" set $A$ is one such that every vertex of $G$ is at distance $\leq 1$ to a vertex in $A$). Now knowing this, Google returns a concept closely related to my $\psi(G)$, namely the domatic number of a graph, which is the maximum number of pairwise disjoint dominating sets. It therefore seems logical to call my $\psi(G)$ the "total domatic number", and some authors seem to have done so.