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 \subseteq 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. García-Vázquez, 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 \subseteq A(D) $ with $ |S| \le 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.