NP-completeness Proof: RBCDN Reduction Problem

Suppose {R1, . . . , Rk} is the set of all possible regions [?] of graph G. Consider a k-dimensional vector C whose i-th entry, C[i], indicates the number of connected components in which G decomposes when all nodes in Ri fails. Then, region-based component decomposition number (RBCDN) of graphG with region R is defined as αR(G) = max1≤i≤k C[i]. Suppose the RBCDN of G with region R is αR(G). If αR(G) is considered to be too high for the application and it requires RBCDN of the network not to exceed αR(G) −K, for some integer K. Assuming each additional link li that can be added to the network has a cost c(i) associated with it, find the least cost link augmentation to the network so that its RBCDN is reduced from αR(G) to αR(G)−K. Formal description of the decision version of this problem is given below.