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A Parameterized Complexity View on Collapsing k-Cores
Luo, Junjie; Molter, Hendrik; Suchý, Ondřej
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.
