On the restricted arc-connectivity of s-geodetic digraphs

For a strongly connected digraph D the restricted arc-connectivity λ′(D) is defined as the minimum cardinality of an arc-cut over all arc-cuts S satisfying that D - S has a non-trivial strong component D ₁ such that D-V (D ₁) contains an arc. Let S be a subset of vertices of D. We denote by ω ⁺(S) the set of arcs uv with u ∈ S and v ∉ S, and by ω ⁻(S) the set of arcs uv with u ∉ S and v ∈ S. A digraph D = (V,A) is said to be λ′-optimal if λ′(D) = ξ′(D), where ξ′(D) is the minimum arc-degree of D defined as ξ(D) = min{ξ′(xy): xy ∈ A}, and ξ′(xy) = min{|ω ⁺({x, y})|, |ω ⁻({x, y})|, |ω ⁺(x) ∪ ω ⁻(y)|, |ω ^-(x)∪ω⁺(y)|}. In this paper a sufficient condition for a s-geodetic strongly connected digraph D to be λ′-optimal is given in terms of its diameter. Furthermore we see that the h-iterated line digraph L ^h (D) of a s-geodetic digraph is λ′-optimal for certain iteration h.