The 1, 2-good-neighbor conditional diagnosabilities of regular graphs

Abstract

Fault diagnosis of systems is an important area of study in the design and maintenance of multiprocessor systems. In 2012, Peng et al. proposed a new measure for the fault diagnosis of systems, namely g-good-neighbor conditional diagnosability, which requires that any fault-free vertex has at least g fault-free neighbors in the system. The g-good-neighbor conditional diagnosabilities of a graph G under the PMC model and the MM* model are denoted by tgPMC(G) and tgMM*(G), respectively. In this paper, we first determine that tgPMC(G)=tgMM*(G) if g ≥ 2. Second, we establish a general result on the 1, 2-good-neighbor conditional diagnosabilities of some regular graphs. As applications, the 1, 2-good-neighbor conditional diagnosabilities of BC graphs, folded hypercubes and four classes of Cayley graphs, namely unicyclic-transposition graphs, wheel-transposition graphs, complete-transposition graphs and tree-transposition graphs, are determined under the PMC model and the MM* model. In addition, we determine the R2-connectivities of BC graphs and folded hypercubes.