Fault diagnosability of regular graphs

Abstract

An interconnection network’s diagnosability is an important measure of its selfdiagnostic capability. In 2012, Peng et al. proposed a measure for fault diagnosis of the network, namely, the h-good-neighbor conditional diagnosability, which requires that every fault-free node has at least h fault-free neighbors. There are two well-known diagnostic models, PMC model and MM* model. The h-goodneighbor diagnosability under the PMC (resp. MM*) model of a graph G, denoted by tPMC h (G) (resp. t MM∗ h (G)), is the maximum value of t such that G is h-good-neighbor t-diagnosable under the PMC (resp. MM*) model. In this paper, we study the 2-good-neighbor diagnosability of some general k-regular kconnected graphs G under the PMC model and the MM* model. The main result tPMC 2 (G) = t MM∗ 2 (G) = g(k − 1)− 1 with some acceptable conditions is obtained, where g is the girth of G. Furthermore, the following new results under the two models are obtained: tPMC 2 (HSn) = t MM∗ 2 (HSn) = 4n− 5 for the hierarchical star network HSn, t PMC 2 (S 2 n) = t MM∗ 2 (S 2 n) = 6n− 13 for the split-star networks S2 n and tPMC 2 (Γn(∆)) = t MM∗ 2 (Γn(∆)) = 6n − 16 for the Cayley graph generated by the 2-tree Γn(∆).