A $t$-fold dominating set in a simple undirected graph $G$ with vertex set $V$ is a subset $D\subseteq V$ such that each vertex of $V\setminus D$ has at least $t$ neighbours in $D$.
I am interested in the counterpart: a (proper) subset $M\subset V$ such that each vertex not in $M$ has at most $t$ neighbours in $M$. Is there a name for such subsets? If there is, could you please give me some references to papers dealing with this type of questions?
I am studying such sets in order to produce subgraphs of a graph with high minimal degree.
I only know that if $D\subset V$ has the property that each vertex not in $D$ has exactly $t$ neighbours in $D$ are called $t$-fold perfect dominating sets. (Leastwise for $t=1$, it is quite common to call them perfect dominating sets.)