A note on the restricted arc connectivity of oriented graphs of girth four

Abstract

Let D be a strongly connected digraph. An arc set S of D is a restricted arc-cut of D if D − S has a non-trivial strong component D1 such that D − V (D1) contains an arc. The restricted arc-connectivity λ(D) of a digraph D is the minimum cardinality over all restricted arc-cuts of D. A strongly connected digraph D is λ-connected when λ(D) exists. This paper presents a family F of strong digraphs of girth four that are not λ-connected and for every strong digraph D /∈ F with girth four it follows that it is λ-connected. Also, an upper and lower bound for λ(D) are given.