On the restricted arc-connectivity of s-geodetic digraphs

Abstract

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 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. Furthermore we see that the h-iterated line digraph $L^h$(D) of a s-geodetic digraph is λ′-optimal for certain iteration h.Postprint (published version

On the optimality of 3-restricted arc connectivity for digraphs and bipartite digraphs

“…In [11], Hellwig and Volkmann concluded many sufficient conditions for digraphs to be λ-optimal. Besides, sufficient conditions for digraphs to be λ ′ -optimal were also given by several authors, for example by Balbuena et al [1][2][3][4], Chen et al [5,6], Grüter and Guo [7,8], Liu and Zhang [9], Volkmann [12] and Wang and Lin [13]. However, closely related conditions for λ 3 -optimal digraphs have received little attention until recently.…”

λ′-Optimality of Bipartite Digraphs

“…However, closely related results on restricted arc-connectivity have not received attention until recently. In [3,6,12], the authors study the restricted arc-connectivity of s-geodetic digraphs, Cartesian product digraphs and generalized tournaments respectively. In [4], the authors introduce the concept of super-λ digraphs and provide a sufficient condition for an s-geodetic digraph to be super-λ .…”

On the super‐restricted arc‐connectivity of s ‐geodetic digraphs

“…Some sufficient conditions for some families of digraphs to attain λ′ ‐optimality can be found in Refs. [8, 14, 30].…”

“…Let D be a λ′ ‐ connected s ‐geodetic digraph and \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}\omega^+(F)=\lbrack X,\overline{X}\rbrack \end{align*}\end{document} a λ′ ‐ cut. If λ′( D ) <ξ ′( D ) then there exists some vertex u ∈ F such that d ( u , X ) ≥ s ‐ 1 and there exists some vertex \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}\overline{u}\in \overline{F}\end{align*}\end{document} such that \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}d(\overline{X},\overline{u})\ge s-1\end{align*}\end{document} [8].…”