# Vertex Cover Problem Parameterized Above and Below Tight Bounds

## TL;DR

This paper investigates the parameterized complexity of the Vertex Cover problem relative to tight bounds, identifying which parameterizations are fixed-parameter tractable and which are W[1]-hard or W[2]-hard.

## Contribution

It proves that two parameterizations are fixed-parameter tractable while two others are W[1]-hard or W[2]-hard, clarifying the complexity landscape of the problem.

## Key findings

- Two parameterizations are fixed-parameter tractable.
- Two parameterizations are W[1]-hard or W[2]-hard.
- Provides complexity classification for parameterizations above and below tight bounds.

## Abstract

We study the well-known Vertex Cover problem parameterized above and below tight bounds. We show that two of the parameterizations (both were suggested by Mahajan, Raman and Sikdar, J. Computer and System Sciences, 75(2):137--153, 2009) are fixed-parameter tractable and two other parameterizations are W[1]-hard (one of them is, in fact, W[2]-hard).