Cyclic arc-connectivity in a Cartesian product digraph

A digraph D is cycle separable if it contains two vertex disjoint directed cycles. For a cycle separating digraph D , an arc set S is a cycle separating arc-cut if D − S has at least two strong components containing directed cycles. The cyclic arc-connectivity λ c ( D ) is the minimum cardinality of all cycle separating arc-cuts. In this work, we study λ c ( D ) for the Cartesian product digraph D = D 1 × D 2 . We give a necessary and sufficient condition for D 1 × D 2 to be cycle separable, and show that λ c ( D 1 × D 2 ) = 0 if D 1 × D 2 is cycle separable but not strongly connected. For the case where D = D 1 × D 2 is strongly connected, we give an upper bound and a lower bound for λ c ( D ) . In particular, it can be determined that λ c ( C n 1 × C n 2 × ⋯ × C n k ) = ( k − 1 ) min { n 1 , n 2 , … , n k } , where C n i is a directed cycle of length n i .