On cyclic vertex-connectivity of Cartesian product digraphs

Abstract

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 κ _c (D) is the minimum cardinality of all cyclic vertex-cuts of D.

In this paper, we study κ _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 κ _c (D). Furthermore, the exact value of $\kappa_{c}(C_{n_{1}}\times C_{n_{2}}\times\cdots\times C_{n_{k}})$ is determined, where $C_{n_{i}}$ is the directed cycle of length n _i for i=1,2,…,k.