The g-faulty-block connectivity of folded hypercubes

Abstract

There are some attacks on the network, such as botnet attack, DDoS attack and Local Area Network Denial attack, which are attacked on certain group of clustered nodes in the network. At present, the existing connectivity has certain defects in reflecting the fault-tolerant ability of the network under these network attacks. In order to measure the fault tolerance and reliability of a network which is attacked on certain group of clustered nodes in the network by attacker, Lin et al. (IEEE Trans Comput 70:1719–1731, 2021) proposed the g-faulty-block connectivity. A subset $F\subseteq V(G)$ is called a g-faulty-block of a graph G if $G-F$ is disconnected, each component of it has at least $g+1$ vertices and the subgraph induced by F is connected. The cardinality of a minimum g-faulty-block of G, denoted by $\text{FB}\kappa _g(G)$, is the g-faulty-block connectivity of G. Larger h-fault block connectivity means that an attacker must launch an attack on a larger block of connected nodes so that each remaining component is not too small, which in turn limits the size of the larger components. The larger the h-fault block, the more difficult it is for an attacker to accomplish this goal. In this paper, we obtain $\text{FB}\kappa _0(\text{FQ}_n)=2n+1$, $\text{FB}\kappa _1(\text{FQ}_n)=3n-1$ and $\text{FB}\kappa _g(\text{FQ}_n)=(g+2)n-3g+3$ for $2\le g\le n-4$ and $n\ge 7$, where $\text{FQ}_n$ is n-dimension folded hypercube.