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. 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 $n\geq 3$ (Lin, C.-K., Zhang, L.-L., Fan, J.-X. and Wang, D.-J. (2016) Structure connectivity and substructure connectivity of hypercubes. Theor. Comput. Sci., 634, 97–107). Sabir et al. 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 $n\geq 7$ (Sabir, E. and Meng, J.(2018) Structure fault tolerance of hypercubes and folded hypercubes. Theor. Comput. Sci., 711, 44–55). 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.

中文翻译：

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超立方体和折叠超立方体的星-结构连通性和星-子结构连通性

作为顶点连通性的推广，对于连通图 $G$ 和 $T$，$T$-结构连通性 $\kappa (G; T)$ (resp. $T$-substructure connectivity $\kappa ^{s} $G$ 的 (G; T)$) 是 $G$ 的一组子图 $F$ 的最小基数，每个子图都同构于 $T$（对应于 $T$ 的连通子图），因此 $ GF$ 已断开连接。对于 $n$ 维超立方体 $Q_{n}$，Lin 等人。显示 $\kappa (Q_{n};K_{1,1})=\kappa ^{s}(Q_{n};K_{1,1})=n-1$ 和 $\kappa (Q_{n };K_{1,r})=\kappa ^{s}(Q_{n};K_{1,r})=\lceil \frac{n}{2}\rceil $ for $2\leq r\leq 3$ 和 $n\geq 3$ (Lin, C.-K., Zhang, L.-L., Fan, J.-X. 和 Wang, D.-J. (2016) 结构连通性和子结构连通性超立方体. 理论. 计算机科学., 634, 97–107). 萨比尔等人。得到$\kappa (Q_{n};K_{1,4})=\kappa ^{s}(Q_{n};K_{1, 4})= \lceil \frac{n}{2}\rceil $ 对于 $n\geq 6$ 和 $n$ 维折叠超立方体 $FQ_{n}$, $\kappa (FQ_{n};K_ {1,1})=\kappa ^{s}(FQ_{n};K_{1,1})=n$, $\kappa (FQ_{n};K_{1,r})=\kappa ^ {s}(FQ_{n};K_{1,r})= \lceil \frac{n+1}{2}\rceil $ 与 $2\leq r\leq 3$ 和 $n\geq 7$ (Sabir , E. and Meng, J.(2018) Structure fault tolerance of hypercubes and folded hypercubes. Theor. Comput. Sci., 711, 44-55)。他们提出了一个开放问题，即确定一般 $r$ 的 $Q_n$ 和 $FQ_n$ 的 $K_{1,r}$-结构连通性。在本文中，我们得到对于每个整数 $r\geq 2$，$\kappa (Q_{n};K_{1,r})$ $=\kappa ^{s}(Q_{n};K_{ 1,r})$ $=\lceil \frac{n}{2}\rceil $ 和 $\kappa (FQ_{n};K_{1,r})=\kappa ^{s}(FQ_{n} ;K_{1,r})= \lceil \frac{n+1}{2}\rceil $ 用于所有大于 $r$ 的整数 $n$ 在 quare scale 中。对于 $4\leq r\leq 6$，