The g-Extra Conditional Diagnosability of Graphs in Terms of g-Extra Connectivity

The [Formula: see text]-extra conditional diagnosability and [Formula: see text]-extra connectivity are two important parameters to measure ability of diagnosing faulty processors and fault tolerance in a multiprocessor system. The [Formula: see text]-extra conditional diagnosability [Formula: see text] of graph [Formula: see text] is defined as the diagnosability of a multiprocessor system under the assumption that every fault-free component contains more than [Formula: see text] vertices. While the [Formula: see text]-extra connectivity [Formula: see text] of graph [Formula: see text] is the minimum number [Formula: see text] for which there is a vertex cut [Formula: see text] with [Formula: see text] such that every component of [Formula: see text] has more than [Formula: see text] vertices. In this paper, we study the [Formula: see text]-extra conditional diagnosability of graph [Formula: see text] in terms of its [Formula: see text]-extra connectivity, and show that [Formula: see text] under the MM* model with some acceptable conditions. As applications, the [Formula: see text]-extra conditional diagnosability is determined for some BC networks such as hypercubes, varietal hypercubes, and [Formula: see text]-ary [Formula: see text]-cubes under the MM* model.

Two-Round Diagnosability Measures for Multiprocessor Systems

In a multiprocessor system, as a key measure index for evaluating its reliability, diagnosability has attracted lots of attentions. Traditional diagnosability and conditional diagnosability have already been widely discussed. However, the existing diagnosability measures are not sufficiently comprehensive to address a large number of faulty nodes in a system. This article introduces a novel concept of diagnosability, called two-round diagnosability, which means that all faulty nodes can be identified by at most a one-round replacement (repairing the faulty nodes). The characterization of two-round t-diagnosable systems is provided; moreover, several important properties are also presented. Based on the abovementioned theories, for the n-dimensional hypercube Qn, we show that its two-round diagnosability is n2+n/2, which is n+1/2 times its classic diagnosability. Furthermore, a fault diagnosis algorithm is proposed to identify each node in the system under the PMC model. For Qn, we prove that the proposed algorithm is the time complexity of On2n.

Reliability of DQcube Based on g-Extra Conditional Fault

Diagnosability and connectivity are important metrics for the reliability and fault diagnosis capability of interconnection networks, respectively. The g-extra connectivity of a graph G, denoted by $\kappa _g(G)$, is the minimum number of vertices whose deletion will disconnect the network and every remaining component has more than $g$ vertices. The g-extra conditional diagnosability of graph G, denoted by $t_g(G)$, is the maximum number of faulty vertices that the graph G can guarantee to identify under the condition that every fault-free component contains at least g+1 vertices. In this paper, we first determine that g-extra connectivity of DQcube is $\kappa _g(G)=(g+1)(n+1)-\frac{g(g+3)}{2}$ for $0\leq g\leq n-3$ and then show that the g-extra conditional diagnosability of DQcube under the PMC model $(n\geq 4, 1\leq g\leq n-3)$ and the MM$^\ast$ model $(n\geq 7, 1\leq g\leq \frac{n-3}{4})$ is $t_g(G)=(g+1)(n+1)-\frac{g(g+3)}{2}+g$, respectively.