Restricted arc-connectivity of digraphs

Abstract

Since interconnection networks are often modeled by graphs or digraphs, the edge-connectivity of a graph or arc-connectivity of a digraph are important measurements for fault tolerance of networks. The restricted edge-connectivity λ ′ ( G ) of a graph G is the minimum cardinality over all edge-cuts S in a graph G such that there are no isolated vertices in G − S . A connected graph G is called λ ′ -connected, if λ ′ ( G ) exists. In 1988, Esfahanian and Hakimi [A.H. Esfahanian, S.L. Hakimi, On computing a conditional edge-connectivity of a graph, Inform. Process. Lett. 27 (1988), 195–199] have shown that each connected graph G of order n ⩾ 4 , except a star, is λ ′ -connected and satisfies λ ′ ( G ) ⩽ ξ ( G ) , where ξ ( G ) is the minimum edge-degree of G. If D is a strongly connected digraph, then we call in this paper an arc set S a restricted arc-cut of D if D − S has a non-trivial strong component D 1 such that D − V ( D 1 ) contains an arc. The restricted arc-connectivity λ ′ ( D ) is the minimum cardinality over all restricted arc-cuts S. We observe that the recognition problem, whether λ ′ ( D ) exists for a strongly connected digraph D is solvable in polynomial time. Furthermore, we present some analogous results to the above mentioned theorem of Esfahanian and Hakimi for digraphs, and we show that this theorem follows easily from one of our results.