Parameterized complexity of critical node cuts

We consider the following 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(κ)⋅n^O(1) time (i.e., whether or not it is fixed-parameter tractable), for various natural parameters κ. 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 κ^O(1), where κ is the given parameter.