3-Restricted arc connectivity of digraphs

Abstract

The k-restricted arc connectivity of digraphs is a common generalization of the arc connectivity and the restricted arc connectivity. An arc subset S of a strong digraph D is a k-restricted arc cut if D−S has a strong component D′ with order at least k such that D−V(D′) contains a connected subdigraph with order at least k. The k-restricted arc connectivity λk(D) of a digraph D is the minimum cardinality over all k-restricted arc cuts of D.Let D be a strong digraph with order n≥6 and minimum degree δ(D). In this paper, we first show that λ3(D) exists if δ(D)≥3 and, furthermore, λ3(D)≤ξ3(D) if δ(D)≥4, where ξ3(D) is the minimum 3-degree of D. Next, we prove that λ3(D)=ξ3(D) if δ(D)≥n+32. Finally, we give examples showing that these results are best possible in some sense.