An Interplay between Critical Node Detection and Epidemic Models

For a given graph and a positive number k, the goal of the Critical Node Detection Problem (CNDP) is to find the set of $k$ nodes, named critical nodes, whose removal minimize the connectivity between the surviving nodes. The CNDP has been extensively studied in the literature and is gaining special attention in the vulnerability evaluation of telecommunication networks. More recently, a worst-case analysis of an epidemic model was introduced, where a disease is spread among a given population. The goal is to find a set of nodes to be immunized that minimize the number of dead-nodes as a result. This extremal analysis is captured by a combinatorial optimization problem, called Graph Fragmentation Problem (GFP). In this paper, we show that the CNDP and the GFP are identical combinatorial problems, in the sense that the globally optimal solution is identical under the same instances. As corollary, we conclude universal inapproximability results for the CNDP.