Vertex Cover Structural Parameterization Revisited

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.