Structure and substructure connectivity of folded divide-and-swap cube

Let $H$ be a connected subgraph of a graph $G$. The structure connectivity of $G$, denoted by $\kappa(G;H)$, is the minimum number of a set of connected subgraphs in $G$, whose removal disconnects $G$ and each element in the set is isomorphic to $H$. The substructure connectivity of $G$, denoted by $\kappa^s(G;H)$, is the minimum number of a set of connected subgraphs in $G$, whose removal disconnects $G$ and each element in the set is isomorphic to a connected subgraph of $H$. In this paper, we determine $H$-structure connectivity and $H$-substructure connectivity of folded divide-and-swap cube $FDSC_n$ for $H\in\{K_1, K_{1,1}, K_{1,m} (2\leq m \leq d+1) \}$ where $n=2^d$. We show that $\kappa(FDSC_n;K_1)=\kappa^s(FDSC_n;K_1)=d+2$, $\kappa(FDSC_n;K_{1,1})=\kappa^s(FDSC_n;K_{1,1})=d+1$ for $d\geq1$ and $\kappa(FDSC_n;K_{1,m})=\kappa^s(FDSC_n;K_{1,m})=\lfloor\frac{d}{2}\rfloor+1$ for $d\geq1$ and $2\leq m \leq d+1$.