The non-inclusive g-good-neighbor diagnosability of interconnection networks

Motivated by the definitions of g-good-neighbor diagnosability and non-inclusive diagnosability, we propose a new diagnosability—the non-inclusive g-good-neighbor diagnosability tNg(G) of a multiprocessor system G, which requires every pair of g-good-neighbor faulty sets is non-inclusive. The Rg-conditional diagnosability tRg(G) of a system G is a generalization of conditional diagnosability, which requires at least g fault-free neighbors for each node. In this paper, we explore the relationships between the non-inclusive g-good-neighbor diagnosability and the Rg-conditional diagnosability of G under the PMC and MM* models. We first show tNg(G)≤tRg(G) for g≥1, and also give some conditions for equality. Next, we discuss the non-inclusive g-good-neighbor diagnosability of hypercubes, (n,k)-star graphs and (n,k)-bubble-sort graphs. We show that the non-inclusive g-good-neighbor diagnosability of n-dimensional hypercubes is less that its Rg-conditional diagnosability for 2≤g≤n−22, and determine the non-inclusive g-good-neighbor diagnosability of (n,k)-star graphs and (n,k)-bubble-sort graphs. Finally, we plot and compare the non-inclusive g-good-neighbor diagnosability and the g-good-neighbor diagnosability of (n,k)-star graphs and (n,k)-bubble-sort graphs under the PMC and MM* models, respectively. It can be seen that their non-inclusive g-good-neighbor diagnosability is significantly larger than their g-good-neighbor diagnosability.