2026-10-01T07:51:44Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:26061 2026-09-23T23:59:13Z ddc:004 open_access

Parameterized Critical Node Cut Revisited Knop, Dušan Melissinos, Nikolaos Vasilakis, Manolis Critical Node Cut Parameterized Complexity Treewidth We study how to sparsify connectivity in graphs under a tight deletion budget. Given a graph G and integers k,x ≥ 0, Critical Node Cut (CNC) asks whether we can delete at most k vertices so that the number of remaining unordered pairs of connected vertices is at most x. CNC generalizes Vertex Cover (the case x = 0) and models tasks in network design, epidemiology, and social network analysis. We comprehensively map the structural parameterized complexity landscape for Critical Node Cut. First, we prove W[1]-hardness for the combined parameter k + fes + Δ + pw, where fes is the feedback edge set number, Δ the maximum degree, and pw the pathwidth of the input graph, respectively. This significantly improves over the known W[1]-hardness for k+tw, where tw denotes the treewidth, and is tight in that tree-depth together with maximum degree trivially yields FPT. Second, we give new positive results. Specifically, we identify three structural parameters-max-leaf number, vertex integrity, and modular-width-that render the problem fixed-parameter tractable, and develop a polynomial-time algorithm for graphs of constant clique-width. Third, leveraging a technique introduced by Lampis [ICALP '14], we develop an FPT approximation scheme that, for any ε > 0, computes a (1+ε)-approximate solution in time (tw / ε)^{𝒪(tw)} n^{𝒪(1)}. Finally, we show that CNC admits no polynomial kernel when parameterized by vertex cover number, unless standard assumptions fail. Together, these results substantially sharpen the known complexity landscape for CNC. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Dušan Knop and Nikolaos Melissinos and Manolis Vasilakis 2026 Is Part Of LIPIcs, Volume 370, 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SWAT.2026.25 urn:nbn:de:0030-drops-260617 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2026.25 eng https://creativecommons.org/licenses/by/4.0/legalcode