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 of D is the cardinality of a minimum set of subdigraphs in D, whose removal results in a non-strongly connected digraph and . The T-substructure connectivity of D is the cardinality of a minimum set of subdigraphs in D, whose removal results in a non-strongly connected digraph and each element is isomorphic to a connected subdigraph of T. In this work, we study resp. for , and ; resp. for and ; and resp. for , , and , where is the directed k-ary n-cube, is the in-star on t + 1 vertices, and are, respectively, the directed path and cycle of length t.