The t/k-diagnosability of the BC graphs

Processor fault diagnosis takes an important role in fault-tolerant computing on multiprocessor systems. There are two classical diagnosis strategies - the precise strategy and the pessimistic strategy, both of which are based on the well-known PMC diagnostic model. Nevertheless, the degree of diagnosability of the system is limited under these two strategies. A better method, called the t/k-diagnosis strategy, is proposed by Somani and Peleg, in which the identified fault-set is allowed to contain at most k fault-free processors. Using this diagnosis strategy, the degree of diagnosability of the hypercube increases greatly as the number of the fault-free processors in the fault-set increases. We study the t/k-diagnosability of so-called BC graphs that include hypercubes, crossed cubes, Mobius cubes, and twisted cubes, etc. We show that any n-dimensional BC graph is t(n, k)/k-diagnosable when n≥4 and 0≤k≤n, where t(n,k)=(k+1)n-1/2(k+1)(k+2)+1. Therefore, the crossed cube, the Mobius cube, and the twisted cube all have the same t/k-diagnosability as the hypercube. As a result, the algorithms developed for diagnosis on the hypercube may also be used to diagnose multiprocessor systems whose network topologies are based on BC graphs.