  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. [AAAI 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. We show that Collapsed k-Core is W[1]-hard
when parameterized by b and in FPT when parameterized by (b + x) if k <= 2.
Furthermore, we show 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.
