A novel conditional connectivity to measure network reliability: r-component block connectivity

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. To measure the fault tolerance and reliability of a network which is attacked on certain group of clustered nodes in the network by attackers, we propose the r-component block connectivity. A subset $F\subseteq V(G)$ is called a r-component block cut of a graph G if G−F is disconnected, moreover G−F has at least r components and the subgraph induced by F is connected. The cardinality of a minimum r-component block cut of G, denoted by $\mathrm{CB}{\kappa }_r(G)$, is the r-component block connectivity of G. In this paper, we obtain $\mathrm{CB}{\kappa }_2(Q_n)=2n-1$ and $\mathrm{CB}{\kappa }_{r+1}(Q_n)=(r+1)n-3r+3$ for $2\le r\le n-2$ and $n\ge 4$, where $Q_n$ is a n-dimension hypercube.

KEYWORDS:

• r-component block cut
• r-component block connectivity
• hypercube

MATHEMATICS SUBJECT CLASSIFICATIONS:

• 05C40
• 05C07

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability

The data used to support the findings of this study are available from the corresponding author upon request.

Additional information

Funding

This work is supported by Qinghai University Science Foundation of China (No. 2023-QGY-6), the National Science Foundation of China (Nos. 11661068, 12261074 and 12201335).