As a generalization of vertex connectivity, for connected graphs G and T,
the T-structure connectivity κ(G,T) (resp. T-substructure
connectivity κs(G,T)) of G is the minimum cardinality of a set of
subgraphs F of G that each is isomorphic to T (resp. to a connected
subgraph of T) so that G−F is disconnected. For n-dimensional hypercube
Qn​, Lin et al. [6] showed
κ(Qn​,K1,1​)=κs(Qn​,K1,1​)=n−1 and
κ(Qn​,K1,r​)=κs(Qn​,K1,r​)=⌈2n​⌉ for
2≤r≤3 and n≥3. Sabir et al. [11] obtained that
κ(Qn​,K1,4​)=κs(Qn​,K1,4​)=⌈2n​⌉ for
n≥6, and for n-dimensional folded hypercube FQn​,
κ(FQn​,K1,1​)=κs(FQn​,K1,1​)=n,
κ(FQn​,K1,r​)=κs(FQn​,K1,r​)=⌈2n+1​⌉
with 2≤r≤3 and n≥7. They proposed an open problem of
determining K1,r​-structure connectivity of Qn​ and FQn​ for general
r. In this paper, we obtain that for each integer r≥2,
κ(Qn​;K1,r​)=κs(Qn​;K1,r​)=⌈2n​⌉ and
κ(FQn​;K1,r​)=κs(FQn​;K1,r​)=⌈2n+1​⌉
for all integers n larger than r in quare scale. For 4≤r≤6, we
separately confirm the above result holds for Qn​ in the remaining cases