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

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⊆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 CBκr(G), is the r-component block connectivity of G. In this paper, we obtain CBκ2(Qn)=2n−1 and CBκr+1(Qn)=(r+1)n−3r+3 for 2≤r≤n−2 and n≥4, where Qn is a n-dimension hypercube.