Fault tolerance assessment for hamming graphs based on r-restricted R-structure(substructure) fault pattern

The interconnection network between the storage system and the multi-core computing system is the bridge for communication of enormous amounts of data access and storage, which is the critical factor in affecting the performance of high-performance computing systems. By enforcing additional restrictions on the definition of R-structure and R-substructure connectivities to satisfy that each remaining vertex has not less than r neighbors, we can dynamically assess the cardinality of the separated component to meet the above conditions under structure faulty, thereby enhancing the evaluation of the fault tolerance and reliability of high-performance computing systems. Let R be a connected subgraph of a connected graph G. The r-restricted R-structure connectivity ${\kappa }_r(G;R)$ (resp. r-restricted R-substructure connectivity ${\kappa }_r^s(G;R)$) of G is the minimum cardinality of a set of subgraphs $F=\{F_1,F_2,\dots ,F_m\}$ such that $F_i$ is isomorphic to R (resp. $F_i$ is a connected subgraph of R) for $1\le i\le m$, and $G-F$ is disconnected with the minimum degree of each component being at least r. Note that ${\kappa }_r(G;K_1)$ reduces to r-restricted connectivity ${\kappa }^r\left(G\right)$ (also called r-good neighbor connectivity). In this paper, we focus on ${\kappa }_r(K_L^n;R)$ and ${\kappa }_r^s(K_L^n;R)$ for the L-ary n-dimensional hamming graph $K_L^n$, where $R\in \{K_1,K_{1,1},K_L^1\}$. For $0\le r\le n-3$, $n\ge 3$ and $L\ge 3$, we determine the $(L-1)r$-good neighbor connectivity of $K_L^n$, i.e., ${\kappa }^{(L-1)r}\left(K_L^n\right)=(L-1)(n-r)L^r$, and the $(L-1)r$-good neighbor diagnosability of $K_L^n$ under the PMC model and MM* model, i.e., $t^{(L-1)r}\left(K_L^n\right)=\left[\right(L-1\left)\right(n-r)-1]L^r-1$. And we also drive that ${\kappa }_{(L-1)r}(K_L^n;K_{1,1})={\kappa }_{(L-1)r}^s(K_L^n;K_{1,1})=\frac{1}{2}(L-1)L^r(n-r)$ for $1\le r\le n-3$, $n\ge 4$. Moreover, we offer an upper bound of ${\kappa }_2(K_L^n;K_L^1)$ (resp. ${\kappa }_2^s(K_L^n;K_L^1))$ for $n\ge 3$, and establish that it is sharp for ternary n-cubes $K_3^n$. Specifically, ${\kappa }_2(K_3^n;K_3^1)={\kappa }_2^s(K_3^n;K_3^1)=2(n-1)$ for $n\ge 3$.