The star-structure connectivity and star-substructure connectivity of hypercubes and folded hypercubes
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As a generalization of vertex connectivity, for connected graphs G and T, the T-structure connectivity κ(G, T) (resp. T-substructure connectivity κ^s(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 Q_n, Lin et al. [6] showed κ(Q_n,K_1,1)=κ^s(Q_n,K_1,1)=n-1 and κ(Q_n,K_1,r)=κ^s(Q_n,K_1,r)=⌈n/2⌉ for 2≤ r≤ 3 and n≥ 3. Sabir et al. [11] obtained that κ(Q_n,K_1,4)=κ^s(Q_n,K_1,4)=⌈n/2⌉ for n≥ 6, and for n-dimensional folded hypercube FQ_n, κ(FQ_n,K_1,1)=κ^s(FQ_n,K_1,1)=n, κ(FQ_n,K_1,r)=κ^s(FQ_n,K_1,r)=⌈n+1/2⌉ with 2≤ r≤ 3 and n≥ 7. They proposed an open problem of determining K_1,r-structure connectivity of Q_n and FQ_n for general r. In this paper, we obtain that for each integer r≥ 2, κ(Q_n;K_1,r)=κ^s(Q_n;K_1,r)=⌈n/2⌉ and κ(FQ_n;K_1,r)=κ^s(FQ_n;K_1,r)= ⌈n+1/2⌉ for all integers n larger than r in quare scale. For 4≤ r≤ 6, we separately confirm the above result holds for Q_n in the remaining cases.
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