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 κ(G;H)\documentclass[12pt]{minimal}
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\begin{document}$$\kappa (G;\; H)$$\end{document} (resp. H-substructure connectivity κs(G;H)\documentclass[12pt]{minimal}
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\begin{document}$$\kappa ^{s}(G;\; H)$$\end{document}) of G is the minimum cardinality of a set of subgraphs F\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {F}$$\end{document} of G that each is isomorphic to H (resp. a connected subgraph of H) so that G-F\documentclass[12pt]{minimal}
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\begin{document}$$G- \mathcal {F}$$\end{document} is disconnected or the singleton. In this paper, we compute the star (K1,m\documentclass[12pt]{minimal}
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\begin{document}$$K_{1,m}$$\end{document})-structure connectivity of n-dimensional folded hypercubes FQn\documentclass[12pt]{minimal}
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\begin{document}$$FQ_{n}$$\end{document} and augmented cubes AQn\documentclass[12pt]{minimal}
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\begin{document}$$AQ_{n}$$\end{document}, which are popular variants of n-dimensional hypercubes Qn\documentclass[12pt]{minimal}
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\begin{document}$$Q_n$$\end{document} as attractive interconnection network prototypes for multiple processor systems. By a large component approach, we obtain that κ(FQn;K1,m)=κs(FQn;K1,m)=⌈n+12⌉\documentclass[12pt]{minimal}
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\begin{document}$$\kappa (FQ_{n};\;K_{1,m})=\kappa ^{s}(FQ_{n};\;K_{1,m})=\lceil \frac{n+1}{2}\rceil$$\end{document} for 2⩽m⩽n-1\documentclass[12pt]{minimal}
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\begin{document}$$2\leqslant m\leqslant n-1$$\end{document}, n⩾7\documentclass[12pt]{minimal}
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\begin{document}$$n\geqslant 7$$\end{document} and κ(AQn;K1,m)=κs(AQn;K1,m)=⌈n-12⌉\documentclass[12pt]{minimal}
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\begin{document}$$\kappa (AQ_{n};\;K_{1,m})=\kappa ^{s}(AQ_{n};\;K_{1,m})=\lceil \frac{n-1}{2}\rceil$$\end{document} for 4⩽m⩽3n-154\documentclass[12pt]{minimal}
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\begin{document}$$4\leqslant m\leqslant \frac{3n-15}{4}$$\end{document}, which much improve some known results with very restricted m.
