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As a generalization of vertex connectivity, for connected graphs $G$ and $T$,the $T$-structure connectivity $\kappa(G, T)$ (resp. $T$-substructureconnectivity $\kappa^{s}(G, T)$) of $G$ is the minimum cardinality of a set ofsubgraphs $F$ of $G$ that each is isomorphic to $T$ (resp. to a connectedsubgraph 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 ofdetermining $K_{1,r}$-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 $4\leq r\leq 6$, weseparately confirm the above result holds for $Q_n$ in the remaining cases.

The star-structure connectivity and star-substructure connectivity of hypercubes and folded hypercubes