A Parameterized Complexity View on Collapsing

We study the NP-hard graph problem Collapsed k-Core where, given an undirected graph G and integers b , x , and k , we are asked to remove b vertices such that the k -core of remaining graph, that is, the (uniquely determined) largest induced subgraph with minimum degree k , has size at most x . Collapsed k-Core was introduced by Zhang et al. ( 2017 ) and it is motivated by the study of engagement behavior of users in a social network and measuring the resilience of a network against user drop outs. Collapsed k-Core is a generalization of r-Degenerate Vertex Deletion (which is known to be NP-hard for all r ≥ 0) where, given an undirected graph G and integers b and r , we are asked to remove b vertices such that the remaining graph is r -degenerate, that is, every its subgraph has minimum degree at most r . We investigate the parameterized complexity of Collapsed k-Core with respect to the parameters b , x , and k , and several structural parameters of the input graph. We reveal a dichotomy in the computational complexity of Collapsed k-Core for k ≤ 2 and k ≥ 3. For the latter case it is known that for all x ≥ 0 Collapsed k-Core is W[P]-hard when parameterized by b . For k ≤ 2 we show that Collapsed k-Core is W[1]-hard when parameterized by b and in FPT when parameterized by ( b + x ). Furthermore, we outline that Collapsed k-Core is in FPT when parameterized by the treewidth of the input graph and presumably does not admit a polynomial kernel when parameterized by the vertex cover number of the input graph.