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

ABSTRACTThere 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) $ F⊆V(G)is called a r-component block cut of a graph Gif G−Fis disconnected, moreover G−Fhas at least rcomponents and the subgraph induced by Fis connected. The cardinality of a minimum r-component block cut of G, denoted by $ CB\kappa _r(G) $ CBκr(G), is the r-component block connectivity of G. In this paper, we obtain $ CB\kappa _2(Q_n)=2n-1 $ CBκ2(Qn)=2n−1and $ CB\kappa _{r+1}(Q_n)=(r+1)n-3r+3 $ CBκr+1(Qn)=(r+1)n−3r+3for $ 2\leq r\leq ~n-2 $ 2≤r≤n−2and $ n\geq 4 $ n≥4, where $ Q_n $ Qnis a n-dimension hypercube.