Abstract
A pseudoforest is a graph whose connected components have at most one cycle. Let X be a pseudoforest modulator of graph G, i.e. a vertex subset of G such that \(G-X\) is a pseudoforest. We show that Vertex Cover admits a polynomial kernel being parameterized by the size of the pseudoforest modulator. In other words, we provide a polynomial time algorithm that for an input graph G and integer k, outputs a graph \(G'\) and integer \(k'\), such that \(G'\) has \(\mathcal {O}(|X|^{12})\) vertices and G has a vertex cover of size k if and only if \(G'\) has vertex cover of size \(k'\). We complement our findings by proving that there is no polynomial kernel for Vertex Cover parameterized by the size of a modulator to a mock forest (a graph where no cycles share a vertex) unless \(\text {NP}\subseteq \text {coNP/poly}\). In particular, this also rules out polynomial kernels when parameterized by the size of a modulator to cactus graphs.
Supported by Rigorous Theory of Preprocessing, ERC Advanced Investigator Grant 267959.
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Fomin, F.V., Strømme, T.J.F. (2016). Vertex Cover Structural Parameterization Revisited. In: Heggernes, P. (eds) Graph-Theoretic Concepts in Computer Science. WG 2016. Lecture Notes in Computer Science(), vol 9941. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53536-3_15
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DOI: https://doi.org/10.1007/978-3-662-53536-3_15
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