The Reliability of k-Ary n-Cube Based on Component Connectivity

Abstract

Abstract Connectivity and diagnosability are two crucial subjects for a network’s ability to tolerate and diagnose faulty processors. The $r$-component connectivity $c\kappa _{r}(G)$ of a network $G$ is the minimum number of vertices whose deletion results in a graph with at least $r$ components. The $r$-component diagnosability $ct_{r}(G)$ of a network $G$ is the maximum number of faulty vertices that the system can guarantee to identify under the condition that there exist at least $r$ fault-free components. This paper first establishes that the $(r+1)$-component connectivity of $k$-ary $n$-cube $Q^{k}_{n}$ is $c\kappa _{r+1}(Q^{k}_{n})=-\frac{1}{2}r^{2}+\Big(2n-\frac{1}{2}\Big)r+1$ for $n\geq 2$, $k\geq 4$ and $1\leq r\leq n$. In view of $c\kappa _{r+1}(Q^{k}_{n})$, we prove that the $(r+1)$-component diagnosabilities of $k$-ary $n$-cube $Q^{k}_{n}$ under the PMC model and MM* model are $ct_{r+1}(Q^{k}_{n})=-\frac{1}{2}r^{2}+\Big(2n-\frac{3}{2}\Big)r+2n$ for $n\geq 4$, $k\geq 4$ and $1\leq r\leq n-1$.