# Elimination Distances, Blocking Sets, and Kernels for Vertex Cover

## TL;DR

This paper investigates the kernelization of Vertex Cover through the lens of elimination distances and blocking sets, establishing bounds and conditions for polynomial kernels across various graph classes.

## Contribution

It unifies and generalizes kernelization results for Vertex Cover using blocking sets and elimination distances, providing tight bounds and new polynomial kernelizations.

## Key findings

- Bounded minimal blocking set size is necessary but not sufficient for polynomial kernels.
- Exact maximum size of minimal blocking sets is determined for graphs with bounded elimination distance.
- Polynomial kernels are derived for Vertex Cover parameterized by deletion sets to certain graph classes.

## Abstract

The Vertex Cover problem plays an essential role in the study of polynomial kernelization in parameterized complexity, i.e., the study of provable and efficient preprocessing for NP-hard problems. Motivated by the great variety of positive and negative results for kernelization for Vertex Cover subject to different parameters and graph classes, we seek to unify and generalize them using so-called blocking sets, which have played implicit and explicit roles in many results. We show that in the most-studied setting, parameterized by the size of a deletion set to a specified graph class $\mathcal{C}$, bounded minimal blocking set size is necessary but not sufficient to get a polynomial kernelization. Under mild technical assumptions, bounded minimal blocking set size is showed to allow an essentially tight efficient reduction in the number of connected components. We then determine the exact maximum size of minimal blocking sets for graphs of bounded elimination distance to any hereditary class $\mathcal{C}$, including the case of graphs of bounded treedepth. We get similar but not tight bounds for certain non-hereditary classes $\mathcal{C}$, including the class $\mathcal{C}_{LP}$ of graphs where integral and fractional vertex cover size coincide. These bounds allow us to derive polynomial kernels for Vertex Cover parameterized by the size of a deletion set to graphs of bounded elimination distance to, e.g., forest, bipartite, or $\mathcal{C}_{LP}$ graphs.