Critical node cut parameterized by treewidth and solution size is W[1]-Hard

In the Critical Node Cut problem, given an undirected graph G and two non-negative integers k and μ, the goal is to find a set S of exactly k vertices such that after deleting S we are left with at most μ connected pairs of vertices. Back in 2015, Hermelin et al. studied the aforementioned problem under the framework of parameterized complexity. They considered various natural parameters, namely, the size k of the desired solution, the upper bound μ on the number of remaining connected pairs, the lower bound b on the number of connected pairs to be removed, and the treewidth tw(G) of the input graph G. For all but one combinations of the above parameters, they determined whether Critical Node Cut is fixed-parameter tractable and whether it admits a polynomial kernel. The only question they left open is whether the problem remains fixed-parameter tractable when parameterized by k+ tw(G). We answer this question in the negative via a new problem of independent interest, which we call SumCSP. We believe that SumCSP can be a useful starting point for showing hardness results of the same nature, i.e. when the treewidth of the graph is part of the parameter.