Given a graph $G = (V,E)$, a vertex subset $S \subseteq V$ is called [*$t$-stable*]{} (or [*$t$-dependent*]{}) if the subgraph $G[S]$ induced on $S$ has maximum degree at most $t$. The [*$t$-stability number*]{} $\alpha_t(G)$ of $G$ is the maximum order of a $t$-stable set in $G$. The 