The Star-Structure Connectivity and Star-Substructure Connectivity of Hypercubes and Folded Hypercubes

As a generalization of vertex connectivity, for connected graphs G and T, the T-structure connectivity kappa (G; T) (resp. T-substructure connectivity kappa(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. showed kappa (Q(n); K-1,K-r) = kappa(s)(Q(n); K-1,K-r) = inverted right perpendicularn/2inverted left perpendicular and kappa (Q(n); Ki,r) = kappa(s)(Qn ; Ki,r) = 51 for 2 <= r <= 3 and n >= 3 (Lin, C.-K., Zhang, L.-L., Fan, J.-X. and Wang, D.-J. (2016) Structure connectivity and substructure connectivity of hypercubes. Theor. Comput. Sci., 634, 97-107). Sabir et al. obtained that kappa(Q(n); K-1,K-4) = kappa(s) (Q(n); K-1,K-4) = inverted right perpendicularn/2inverted left perpendicular for n >= 6 and for n-dimensional folded hypercube FQ(n), kappa(FQ(n); K-1,K-1) = kappa(s) (FQ(n); K-1,K-1) = n, kappa(FQ(n); K-1,K-r) = kappa(s)(FQ(n); K-1,K-r) = inverted right perpendicularn+1/2inverted left perpendicular with 2 <= r <= 3 and n >= 7 (Sabir, E. and Meng, J.(2018) Structure fault tolerance of hypercubes and folded hypercubes. Theor. Comput. Sci., 711, 44-55). They proposed an open problem of determining K-1,K- r-structure connectivity of Q(n) and FQ(n) for general r. In this paper, we obtain that for each integer r >= 2, kappa(Q(n); K-1,K-r) = kappa(s)(Q(n); K-1,K-r) = inverted right perpendicularn/2inverted left perpendicular and kappa(FQ(n); K-1,K-r) = kappa(s)(FQ(n); K-1,K-r) = inverted right perpendicularnn+1/2inverted left perpendicular 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.