On the λ′-optimality of s-geodetic digraphs

For a strongly connected digraphD 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 D1 such that D − V (D1) 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. Further we see that the h-iterated line digraph Lh(D) of a s-geodetic digraph is λ′-optimal for certain iteration h.