Connectivity, super connectivity and generalized 3-connectivity of folded divide-and-swap cubes

Let G be a connected graph and $S\subseteq V\left(G\right)$ with $\left|S\right|\ge 2$. A tree T in G is called an S-tree if $S\subseteq V\left(T\right)$. Two S-trees $T_1$ and $T_2$ are internally disjoint if $E\left(T_1\right)\cap E\left(T_2\right)=\varnothing$ and $V\left(T_1\right)\cap V\left(T_2\right)=S$. For an integer $r\ge 2$, the generalized r-connectivity of a graph G, denoted by ${\kappa }_r\left(G\right)$, is defined as ${\kappa }_r\left(G\right)=\min \left\{{\kappa }_G\right(S\left)\right|S\subseteq V\left(G\right)$ and $\left|S\right|=r\}$, where ${\kappa }_G\left(S\right)$ denotes the maximum number of pairwise internally disjoint S-trees in G. The folded divide-and-swap cube, denoted by $FDS{C}_n$, is a variant of the hypercube. $FDS{C}_n$ has better network cost measured by the product of degree and diameter than the hypercube and folded hypercube. Connectivity and super connectivity are two important parameters to evaluate the reliability of an interconnection network. In addition, as a generalization of traditional connectivity, generalized connectivity can more accurately assess the reliability of an interconnection network. In this paper, we first acquire the (edge) connectivity and super (edge) connectivity of $FDS{C}_n$ and then obtain the generalized 3-connectivity of $FDS{C}_n$.