The structure connectivity $\kappa (G;H)$ and substructure connectivity ${\kappa }^s(G;H)$ are important indicators to measure interconnection network's fault tolerance and reliability. The data center network, denoted by $D_{k,n}$, have been proposed for data centers as a server-centric interconnection network structure, which can support millions of servers with high network capacity by only using commodity switches. In this paper, we obtain $\kappa (D_{k,n};S_m)$ and ${\kappa }^s(D_{k,n};S_m)$ when $k\ge 2$, $n\ge 4$ and $1\le m\le n+k-2$. Furthermore, we obtain both $\kappa (D_{k,n};S_{23})$ and ${\kappa }^s(D_{k,n};S_{23})$ for $k\ge 8$ and $n\ge 8$.

Partial domination was proposed in 2017 on the basis of domination theory, which has good practical application background in communication network. Let $G=(V,E)$ be a graph and $\mathcal{F}$ be a family of graphs, a subset $S\subseteq V$ is called an $\mathcal{F}$-isolating set of $G$ if $G[V\mathrm{\setminus }{N}_G[S]]$ does not contain $F$ as a subgraph for all $F\in \mathcal{F}$. The subset $S$ is called an isolating set of $G$ if $\mathcal{F}=\{K_2\}$ and $G[V\mathrm{\setminus }{N}_G[S]]$ does not contain $K_2$ as a subgraph. The isolation number of $G$ is the minimum cardinality of an isolating set of $G$, denoted by $\iota (G)$. The hypercube network and $n$-star network are the basic models for network systems, and many more complex network structures can be built from them. In this study, we obtain the sharp bounds of the isolation numbers of the hypercube network and $n$-star network.

At present, the reliability of interconnection networks of multiprocessing systems has become a hot topic of research concern for parallel computer systems. Conditional connectivity is an important parameter to measure the reliability of an interconnected network. In reality, the failure of one node will inevitably have a negative impact on the surrounding nodes. Often it is the specific structures that fail in an interconnected network. Therefore, we propose two novel kinds of connectivity, called $g$-extra $H$-structure connectivity and $g$-extra $H$-substructure connectivity, to go for a more accurate measure of the reliability of the network. Hypercube network is the most dominant interconnection network topology used by computer systems today, for example, the famous parallel computing systems Cray $T3D$, Cray $T3E$, $IBM$ Blue Gene, etc. are built with it as the interconnection network topology. In this paper, we obtain the results of the $g$-extra $H$-structure connectivity and the $g$-extra $H$-substructure connectivity of the hypercubes when the specific structure is $P_k$ and $g=1$.