  This paper considers a kind of generalized measure $\kappa_s^{(h)}$ of fault
tolerance in the $(n,k)$-star graph $S_{n,k}$ and determines
$\kappa_s^{(h)}(S_{n,k})=n+h(k-2)-1$ for $2 \leqslant k \leqslant n-1$ and
$0\leqslant h \leqslant n-k$, which implies that at least $n+h(k-2)-1$ vertices
of $S_{n,k}$ have to remove to get a disconnected graph that contains no
vertices of degree less than $h$. This result contains some known results such
as Yang et al. [Information Processing Letters, 110 (2010), 1007-1011].
