On cyclic vertex-connectivity of Cartesian product digraphs

For a strongly connected digraph D=(V(D),A(D)), a?vertex-cut S?V(D) is a cyclic vertex-cut of D if D?S has at least two strong components containing directed cycles. The cyclic vertex-connectivity ?? _c (D) is the minimum cardinality of all cyclic vertex-cuts of D. In this paper, we study ?? _c (D) for Cartesian product digraph D=D ₁×D ₂, where D ₁,D ₂ are two strongly connected digraphs. We give an upper bound and a lower bound for ?? _c (D). Furthermore, the exact value of $kappa_{c}(C_{n_{1}}times C_{n_{2}}timescdotstimes C_{n_{k}})$ is determined, where $C_{n_{i}}$ is the directed cycle of length n _i for i=1,2,??,k.