Fault-tolerability of the hypercube and variants with faulty subcubes

Abstract

A faulty vertex may probably affect its neighbors and further causes them being faulty, which makes a subnetwork (structure) fail. Therefore, looking into the effect caused by some structures becoming faulty is meaningful. The connectivity and diagnosability are two important parameters to evaluate the fault-tolerability of networks. The connectivity of a network is the minimum number of vertices whose removal will disconnect the network or trivial. We call the network to be t-diagnosable if the number of faulty vertices does not exceed t and all faulty vertices can be identified without a replacement. Let Qn be the n-dimensional hypercube and EH(s,t) be the exchanged hypercube, which is the variant of hypercube. In this paper, we study connectivity, diagnosability, and 1-good-neighbor conditional diagnosability based on structure faults, respectively. Specifically, we first determine the connectivity (1≤k≤n−1 and n≥3) and diagnosability (1≤k≤n−1 and n≥4) of Qn−Qk under the PMC model. Then, we determine the connectivity (2≤k≤min⁡{s,t}) and diagnosability (2≤k≤min⁡{s,t} and min⁡{s,t}≥3) of EH(s,t)−Qk under the PMC model. Finally, we show that the 1-good-neighbor conditional diagnosability of Qn−Qk is 2n−3 for n≥5 and 1≤k≤n−1 under the PMC model, which is almost twice as the traditional diagnosability for a large n.