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 $1\leq r\leq n-1$.

中文翻译：

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基于分量连通性的k-Ary n-Cube的可靠性

连接性和可诊断性是网络容忍和诊断故障处理器能力的两个关键主题。网络$G$ 的$r$-分量连通性$c\kappa _{r}(G)$ 是其删除导致图具有至少$r$ 分量的最小顶点数。一个网络$G$的$r$-分量可诊断性$ct_{r}(G)$是在至少存在$r$无故障的情况下系统可以保证识别的故障顶点的最大数量成分。本文首先确定$k$-ary $n$-cube $Q^{k}_{n}$的$(r+1)$-分量连通性为$c\kappa_{r+1}( Q^{k}_{n})=-\frac{1}{2}r^{2}+\Big(2n-\frac{1}{2}\Big)r+1$ for $n\ geq 2$、$k\geq 4$ 和 $1\leq r\leq n$。鉴于$c\kappa_{r+1}(Q^{k}_{n})$，