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

As a generalization of vertex connectivity, for connected graphs $G$ and $T$, the $T$-structure connectivity $κ(G, T)$ (resp. $T$-substructure connectivity $κ^{s}(G, T)$) 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 $κ(Q_{n},K_{1,1})=κ^{s}(Q_{n},K_{1,1})=n-1$ and $κ(Q_{n},K_{1,r})=κ^{s}(Q_{n},K_{1,r})=\lceil\frac{n}{2}\rceil$ for $2\leq r\leq 3$ and $n\geq 3$. Sabir et al. [11] obtained that $κ(Q_{n},K_{1,4})=κ^{s}(Q_{n},K_{1,4})=\lceil\frac{n}{2}\rceil$ for $n\geq 6$, and for $n$-dimensional folded hypercube $FQ_{n}$, $κ(FQ_{n},K_{1,1})=κ^{s}(FQ_{n},K_{1,1})=n$, $κ(FQ_{n},K_{1,r})=κ^{s}(FQ_{n},K_{1,r})=\lceil\frac{n+1}{2}\rceil$ 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 $r\geq 2$, $κ(Q_{n};K_{1,r})=κ^{s}(Q_{n};K_{1,r})=\lceil\frac{n}{2}\rceil$ and $κ(FQ_{n};K_{1,r})=κ^{s}(FQ_{n};K_{1,r})= \lceil\frac{n+1}{2}\rceil$ 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.