Vertex Cover Structural Parameterization Revisited

TL;DR

This paper demonstrates that Vertex Cover admits a polynomial kernel when parameterized by a pseudoforest modulator, but not when parameterized by a modulator to a mock forest, highlighting the nuanced complexity of structural parameters.

Contribution

It provides a polynomial kernel for Vertex Cover with pseudoforest modulators and establishes kernelization lower bounds for other related parameters.

Findings

Polynomial kernel for Vertex Cover with pseudoforest modulator

No polynomial kernel for modulator to mock forest unless NP ⊆ coNP/poly

Kernelization bounds extend to outerplanar and cactus graphs

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 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 NP is a subset of coNP/poly. In particular, this also rules out polynomial kernels when parameterized by the size of a modulator to outerplanar and cactus graphs.