Parameterized Complexity of Critical Node Cuts

We consider the following natural graph cut problem called Critical Node Cut (CNC): Given a graph G on n vertices, and two positive integers k and x, determine whether G has a set of k vertices whose removal leaves G with at most x connected pairs of vertices. We analyze this problem in the framework of parameterized complexity. That is, we are interested in whether or not this problem is solvable in f(kappa) · nO(1) time (i.e., whether or not it is fixed-parameter tractable), for various natural parameters kappa. We consider four such parameters: - The size k of the required cut. - The upper bound x on the number of remaining connected pairs. - The lower bound y on the number of connected pairs to be removed. - The treewidth w of G. We determine whether or not CNC is fixed-parameter tractable for each of these parameters. We determine this also for all possible aggregations of these four parameters, apart from w+k. Moreover, we also determine whether or not CNC admits a polynomial kernel for all these parameterizations. That is, whether or not there is an algorithm that reduces each instance of CNC in polynomial time to an equivalent instance of size kappaO(1), where kappa is the given parameter.