  As a generalization of vertex connectivity, for connected graphs $G$ and $T$,
the $T$-structure connectivity $\kappa(G, T)$ (resp. $T$-substructure
connectivity $\kappa^{s}(G, T)$) of $G$ is the minimum cardinality of a set of
subgraphs $F$ of $G$ that each is isomorphic to $T$ (resp. to a connected
subgraph of $T$) so that $G-F$ is disconnected. For $n$-dimensional hypercube
$Q_{n}$, Lin et al. [6] showed
$\kappa(Q_{n},K_{1,1})=\kappa^{s}(Q_{n},K_{1,1})=n-1$ and
$\kappa(Q_{n},K_{1,r})=\kappa^{s}(Q_{n},K_{1,r})=\lceil\frac{n}{2}\rceil$ for
$2\leq r\leq 3$ and $n\geq 3$. Sabir et al. [11] obtained that
$\kappa(Q_{n},K_{1,4})=\kappa^{s}(Q_{n},K_{1,4})=\lceil\frac{n}{2}\rceil$ for
$n\geq 6$, and for $n$-dimensional folded hypercube $FQ_{n}$,
$\kappa(FQ_{n},K_{1,1})=\kappa^{s}(FQ_{n},K_{1,1})=n$,
$\kappa(FQ_{n},K_{1,r})=\kappa^{s}(FQ_{n},K_{1,r})=\lceil\frac{n+1}{2}\rceil$
with $2\leq r\leq 3$ and $n\geq 7$. They proposed an open problem of
determining $K_{1,r}$-structure connectivity of $Q_n$ and $FQ_n$ for general
$r$. In this paper, we obtain that for each integer $r\geq 2$,
$\kappa(Q_{n};K_{1,r})=\kappa^{s}(Q_{n};K_{1,r})=\lceil\frac{n}{2}\rceil$ and
$\kappa(FQ_{n};K_{1,r})=\kappa^{s}(FQ_{n};K_{1,r})= \lceil\frac{n+1}{2}\rceil$
for all integers $n$ larger than $r$ in quare scale. For $4\leq r\leq 6$, we
separately confirm the above result holds for $Q_n$ in the remaining cases.
