Fault-Tolerant Strong Menger (Edge) Connectivity of DCC Linear Congruential Graphs

Abstract

With the rapid development and advances of very large scale integration technology and wafer-scale integration technology, multiprocessor systems, taking interconnection networks as underlying topologies, have been widely designed and used in big data era. The topology of an interconnection network is usually represented as a graph. If any two distinct vertices [Formula: see text] in a connected graph [Formula: see text] are connected by min[Formula: see text] vertex (edge)-disjoint paths, then [Formula: see text] is called strongly Menger (edge) connected. In 1996, Opatrny et al. [16] introduced the DCC (Disjoint Consecutive Cycle) linear congruential graph, which consists of [Formula: see text] nodes and is generated by a set of linear functions [Formula: see text] with special properties. In this work, we investigate the strong Menger connectivity of the DCC linear congruential graph [Formula: see text] with faulty vertices or edges, where [Formula: see text], [Formula: see text], gcd[Formula: see text] and [Formula: see text] is a multiple of [Formula: see text]. In detail, we show that [Formula: see text] is strongly Menger connected if [Formula: see text] for any [Formula: see text]. Moreover, we determine that [Formula: see text] is strongly Menger edge connected if [Formula: see text] for any [Formula: see text]. Furthermore, we prove that, under the restricted condition [Formula: see text], [Formula: see text] is strongly Menger edge connected if [Formula: see text] and [Formula: see text] for any [Formula: see text]. In addition, we present some empirical examples to show that the bounds are all optimal in the sense of the maximum number of tolerable edge faults.