Arc fault tolerance of Cartesian product of regular digraphs on super-restricted arc-connectivity.

Let D = (V(D), A(D)) be a strongly connected digraph. An arc set S subset of A(D) is 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 lambda'(D) is the minimum cardinality over all restricted arc-cuts of D. In [C. Balbuena, P. Garcia-Vazquez, A. Hansberg and L.P. Montejano, On the super-restricted arc-connectivity of s-geodetic digraphs, Networks 61 (2013) 20-28], Balbuena et al. introduced the concept of super-lambda' digraphs. In this paper, we first introduce the concept of the arc fault tolerance of a digraph D on the super-lambda' property. We define a super-lambda' digraph D to be m-super-lambda' if D - S is still super-lambda' for any S subset of A(D) with |S| <= m. The maximum value of such m, denoted by S-lambda'(D), is said to be the arc fault tolerance of D on the super-lambda' property. S-lambda'(D) is an index to measure the reliability of networks. Next we provide a necessary and sufficient condition for the Cartesian product of regular digraphs to be super-lambda'. Finally, we give the lower and upper bounds on S-lambda'(D) for the Cartesian product D of regular digraphs and give an example to show that the lower and upper bounds are best possible. In particular, the exact value of S-lambda'(D) is obtained in special cases.