Star-structure connectivity of folded hypercubes and augmented cubes

The connectivity is an important parameter to evaluate the fault-tolerance of a network. As a generalization, structure connectivity and substructure connectivity of graphs were proposed. For connected graphs G and H, the H-structure connectivity $\kappa (G;H)$ (resp. H-substructure connectivity ${\kappa }^s(G;H)$) of G is the minimum cardinality of a set of subgraphs $\mathcal{F}$ of G that each is isomorphic to H (resp. a connected subgraph of H) so that $G-\mathcal{F}$ is disconnected or the singleton. In this paper, we compute the star ($K_{1,m}$)-structure connectivity of n-dimensional folded hypercubes $F{Q}_n$ and augmented cubes $A{Q}_n$, which are popular variants of n-dimensional hypercubes $Q_n$ as attractive interconnection network prototypes for multiple processor systems. By a large component approach, we obtain that $\kappa (F{Q}_n;K_{1,m})={\kappa }^s(F{Q}_n;K_{1,m})=\lceil \frac{n+1}{2}\rceil$ for $2⩽m⩽n-1$, $n⩾7$ and $\kappa (A{Q}_n;K_{1,m})={\kappa }^s(A{Q}_n;K_{1,m})=\lceil \frac{n-1}{2}\rceil$ for $4⩽m⩽\frac{3n-15}{4}$, which much improve some known results with very restricted m.