λ′λ′-Optimality of Bipartite Digraphs

Since the underlying topology of interconnection networks are often modeled as graphs or digraphs, the connectivity and the edge(arc)-connectivity of a digraph are used to measure the reliability of networks. Restricted arc-connectivity is a more refined network reliability index than arc-connectivity.In 2007, Lutz Volkmann [L. Volkmann, Restricted arc-connectivity of digraphs, Inform. Process. Lett. 103 (2007) 234–239] introduced the concept of restricted arc-connectivity to digraphs. In 2008, Shiying Wang and Shangwei Lin [S.Y. Wang, S.W. Lin, λ′λ′-Optimal digraphs, Inform. Process. Lett. 108 (2007) 386–389] introduced the concept of minimum arc-degree and λ′λ′-optimality of digraphs. We call a strongly connected digraph a λ′λ′-optimal digraph if its restricted arc-connectivity is equal to its minimum arc-degree.In this paper, we study the restricted arc-connectivity of bipartite digraphs and give some sufficient conditions for a bipartite digraph to be λ′λ′-optimal.

► We have that for bipartite digraphs the minimum degree to be δ+⩾2δ+⩾2 or δ−⩾2δ−⩾2.
► We give a sufficient condition for a bipartite digraph to be λ′λ′-optimal by δ.
► If D   is a λ2′-connected with λ′(D)<λ2′(D), we have that D   is λ′λ′-optimal if |N+(u)∩N−(v)|⩾2|N+(u)∩N−(v)|⩾2.