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

Let H be a connected subgraph of a graph G. The H-structure connectivity of G, denoted by kappa(G;H), is the minimum cardinality of a set of connected subgraphs in G, whose removal either disconnects G or reduces it to a trivial graph, where each element in the set is isomorphic to H. The H-substructure connectivity of G, denoted by kappa(s)(G;H), is the minimum cardinality of a set of connected subgraphs in G, whose removal either disconnects G or reduces it to a trivial graph, where each element in the set is isomorphic to a connected subgraph of H. In this paper, we investigate the H-structure connectivity and H-substructure connectivity of folded divide-and-swap cube FDSCn for H is an element of{K-1,K-1,K-1,K-1,K-m(2 <= m <= d+2)} where n=2(d). We show that kappa(FDSCn;K-1)=kappa(s)(FDSCn;K-1)=d+2, kappa(FDSCn;K-1,K-1)=kappa(s)(FDSCn;K-1,K-1)=d+1 for d >= 3 and kappa(FDSCn;K-1,K-m)=kappa(s)(FDSCn;K-1,K-m) = left perpendicular d/2 right perpendicular +1 for d >= 1 and 2 <= m <= d+1. Moreover, we show that kappa(s)(FDSCn;K-1,K-d+2) = left perpendicular d/2 right perpendicular+1 for d >= 1 and we provide a bound for kappa(FDSCn;K-1,K-d+2) when d >= 3.