Structure connectivity and substructure connectivity of the directed k-ary n-cube

Given a strongly connected digraph D and a connected subdigraph T of D, the T-structure connectivity κ(D;T) of D is the cardinality of a minimum set of subdigraphs F={T1,T2,…,Tm} in D, whose removal results in a non-strongly connected digraph and Ti≅T (1≤i≤m). The T-substructure connectivity κs(D;T) of D is the cardinality of a minimum set of subdigraphs F={T1,T2,…,Tm} in D, whose removal results in a non-strongly connected digraph and each element Ti (1≤i≤m) is isomorphic to a connected subdigraph of T. In this work, we study κ(Dnk;K1,t←) (resp. κs(Dnk;K1,t←)) for k≥ 3, n≥ 2 and 1≤t