  The connectivity is an important parameter to evaluate the robustness 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 $F$
of $G$ that each is isomorphic to $H$ (resp. to a connected subgraph of $H$) so
that $G-F$ is disconnected or the singleton. As popular variants of hypercubes,
the $n$-dimensional folded hypercubes $FQ_{n}$ and augmented cubes $AQ_{n}$ are
attractive interconnected network prototypes for multiple processor systems. In
this paper, we obtain that
$\kappa(FQ_{n};K_{1,m})=\kappa^{s}(FQ_{n};K_{1,m})=\lceil\frac{n+1}{2}\rceil$
for $2\leqslant m\leqslant n-1$, $n\geqslant 7$, and
$\kappa(AQ_{n};K_{1,m})=\kappa^{s}(AQ_{n};K_{1,m})=\lceil\frac{n-1}{2}\rceil$
for $4\leqslant m\leqslant \frac{3n-15}{4}$.
