The <i>h</i>-faulty-block connectivity of <i>k</i>-ary <i>n</i>-cubes

Abstract The connectivity of a network is an important indicator for assessing its reliability and fault tolerability. However, currently various kinds of connectivity do not well reflect the network’s fault tolerance when facing certain attacks such as Botnet attacks, DDoS attacks, and Local Area Network Denial attacks. Therefore, Lin et al. (A novel measurement for network reliability. IEEE Trans Comput 2021; 70: 17191731.) proposed a new measurement for network reliability. This measurement method can resist the block attack by taking into account of the dispersity of the remaining nodes. Let $G$ be a network, $C \subset V(G)$, and $G[C]$ be a connected subgraph. Then $C$ is called an $h$-faulty-block of $G$ if $G-C$ is disconnected, and every component of $G-C$ has at least $h+1$ nodes. The minimum cardinality over all $h$-faulty-block of $G$ is called $h$-faulty-block connectivity, denoted by $FB_{k_{h}}(G)$. In this paper, we determine $FB_{k_{h}}(Q_{n}^{k})$ for $k$-ary $n$-cube $Q_{n}^{k}$ ($k\geq 3$), a classic interconnection network. We prove that $FB_{k_{0}}(Q_{n}^{3})=3n-1$, $FB_{k_{1}}(Q_{n}^{3})=5n-4$, and $FB_{k_{2}}(Q_{n}^{3})=7n-9$ for $n\geq 3$. Also, we show that $FB_{k_{0}}(Q_{n}^{k})=4n-1$ for $k\geq 4$ and $n\geq 2$, $FB_{k_{1}}(Q_{n}^{4})=6n-4$ for $n\geq 3$, $FB_{k_{1}}(Q_{n}^{k})=6n-3$ for $k\geq 5$ and $n\geq 3$, $FB_{k_{2}}(Q_{n}^{4})=8n-7$ for $n\geq 4$, $FB_{k_{2}}(Q_{n}^{5})=8n-6$ for $n\geq 4$, and $FB_{k_{2}}(Q_{n}^{k})=8n-5$ for $k\geq 6$ and $n\geq 5$.