Arc Fault Tolerance of Cartesian Product of Regular Digraphs on Super-Restricted Arc-Connectivity

• Main Authors: Zhang Guozhen, Wang Shiying
• Format: Article
• Language: English
• Published: Sciendo 2019-02-01
• Series: Discussiones Mathematicae Graph Theory
• Subjects: fault tolerance; restricted arc-connectivity; super-restricted arc- connectivity; cartesian product; regular digraph; 05c40; 68m15
• Online Access: https://doi.org/10.7151/dmgt.2065

Let D = (V (D),A(D)) be a strongly connected digraph. An arc set S ⊆ A(D) is a restricted arc-cut of D if D − S has a non-trivial strong component D1 such that D − V (D1) contains an arc. The restricted arc- connectivity λ‘(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- λ‘ digraphs. In this paper, we first introduce the concept of the arc fault tolerance of a digraph D on the super- λ‘ property. We define a super- λ′ digraph D to be m-super- λ‘ if D − S is still super- λ‘ for any S ⊆ A(D) with |S| ≤ m. The maximum value of such m, denoted by Sλ’(D), is said to be the arc fault tolerance of D on the super- λ‘ property. Sλ’(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- λ‘. Finally, we give the lower and upper bounds on S λ’(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 Sλ’(D) is obtained in special cases.