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.

我们研究了NP难图问题Collapsed k-Core，其中，给定一个无向图G和整数b、x和k，我们被要求移除b个顶点，使得剩余图的k核，即（唯一确定的）具有最小度数k 的最大诱导子图，其大小最多为x。折叠 k-Core是由 Zhang 等人引入的。（2017），它的动机是研究用户在社交网络中的参与行为并衡量网络对用户退出的弹性。倒塌的 k 核是r-Degenerate Vertex Deletion（对于所有r ≥ 0已知为NP -hard ）的泛化，其中，给定一个无向图G和整数b和r，我们被要求删除b个顶点，使得剩余的图是r -degenerate，即它的每个子图至多具有最小度数r。我们研究了Collapsed k-Core关于参数b、x和k的参数化复杂性，以及输入图的几个结构参数。我们揭示了对于k ≤ 2 和k ≥ 3的Collapsed k-Core的计算复杂度的二分法。对于后一种情况，已知对于所有x ≥ 0 Collapsed k-Core当由b参数化时是W[P] -hard . 对于k ≤ 2，我们表明，当由b参数化时，折叠 k-Core是W[1] -hard，当由 ( b + x )参数化时，在FPT 中。此外，我们概述了Collapsed k-Core在FPT 中 当由输入图的树宽参数化时，并且在由输入图的顶点覆盖数参数化时可能不接受多项式核。