2026-07-19T21:47:33Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:8563 2024-03-06T10:41:31Z ddc:004 open_access

Smaller Parameters for Vertex Cover Kernelization Hols, Eva-Maria C. Kratsch, Stefan Vertex Cover Kernelization Structural Parameterization We revisit the topic of polynomial kernels for Vertex Cover relative to structural parameters. Our starting point is a recent paper due to Fomin and Strømme [WG 2016] who gave a kernel with O(|X|^{12}) vertices when X is a vertex set such that each connected component of G-X contains at most one cycle, i.e., X is a modulator to a pseudoforest. We strongly generalize this result by using modulators to d-quasi-forests, i.e., graphs where each connected component has a feedback vertex set of size at most d, and obtain kernels with O(|X|^{3d+9}) vertices. Our result relies on proving that minimal blocking sets in a d-quasi-forest have size at most d+2. This bound is tight and there is a related lower bound of O(|X|^{d+2-epsilon}) on the bit size of kernels. In fact, we also get bounds for minimal blocking sets of more general graph classes: For d-quasi-bipartite graphs, where each connected component can be made bipartite by deleting at most d vertices, we get the same tight bound of d+2 vertices. For graphs whose connected components each have a vertex cover of cost at most d more than the best fractional vertex cover, which we call d-quasi-integral, we show that minimal blocking sets have size at most 2d+2, which is also tight. Combined with existing randomized polynomial kernelizations this leads to randomized polynomial kernelizations for modulators to d-quasi-bipartite and d-quasi-integral graphs. There are lower bounds of O(|X|^{d+2-epsilon}) and O(|X|^{2d+2-epsilon}) for the bit size of such kernels. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Eva-Maria C. Hols and Stefan Kratsch 2018 Is Part Of LIPIcs, Volume 89, 12th International Symposium on Parameterized and Exact Computation (IPEC 2017) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.IPEC.2017.20 urn:nbn:de:0030-drops-85638 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2017.20 eng https://creativecommons.org/licenses/by/3.0/legalcode