The restricted arc connectivity of Cartesian product digraphs

Abstract

For a strongly connected digraph D = ( V ( D ) , A ( D ) ) , an arc set S ⊆ A ( D ) is a k-restricted arc cut if D − S has a non-trivial strongly connected component D 1 such that D − V ( D 1 ) contains an arc. The restricted arc connectivity λ ′ ( D ) is the minimum cardinality of all restricted arc-cuts. In this paper we prove that Cartesian product digraph D = D 1 × D 2 of two strongly connected digraphs D 1 and D 2 is λ ′ -connected. We also give an upper and lower bound for λ ′ ( D ) , respectively. Furthermore, we obtain that λ ′ ( C m → × C n → ) = min { m , n , 3 } and λ ′ ( C m → × K n → ) = n .