On the super‐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. In this paper we prove that every digraph on at least 4 vertices and of minimum degree at least 2 is λ′ ‐connected and λ′(D) ≤ξ′(D), where ξ′(D) is the minimum arc‐degree of D. Also in this paper we introduce the concept of super‐ λ′ digraphs and provide a sufficient condition for an s ‐geodetic digraph to be super‐ λ′. Further, we show that the h ‐iterated line digraph Lh(D) of an s ‐geodetic digraph is super‐ λ′ for a particular h. © 2012 Wiley Periodicals, Inc. NETWORKS, 2013

λ′-Optimality of Bipartite Digraphs

“…, D t such that there is no arc from D j to D i unless j < i [5]. We call such an ordering an acyclic ordering of the strong components of D. [5].) Every acyclic digraph has a vertex of in-degree zero as well as a vertex of out-degree zero.…”

“…In this paper, we obtain some sufficient conditions for a digraph to be λ -optimal. Notations and definitions not given here can be found in [5].…”

“…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 . In this paper, we obtain some sufficient conditions for a digraph to be λ -optimal.…”

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

“…It was proved in [3,13] that for many λ ′ -connected digraphs, ξ ′ (D) is an upper bound of λ ′ (D). In [13], Wang and Lin introduced the concept of…”

“…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.…”