# Above guarantee parameterization for vertex cover on graphs with maximum degree 4

## TL;DR

This paper introduces a new parameterization for the vertex cover problem on degree-4 graphs, resulting in faster algorithms with improved exponential running times.

## Contribution

It presents an algorithm for vertex cover on degree-4 graphs parameterized by r, achieving a running time of O*(1.6253^r), and derives a faster algorithm for the standard problem.

## Key findings

- Algorithm runs in O*(1.6253^r) time for degree-4 graphs.
- Derived an O*(1.2403^k) algorithm for vertex cover on degree-4 graphs.
- Improved exponential time bounds for vertex cover in specific graph classes.

## Abstract

In the vertex cover problem, the input is a graph $G$ and an integer $k$, and the goal is to decide whether there is a set of vertices $S$ of size at most $k$ such that every edge of $G$ is incident on at least one vertex in $S$. We study the vertex cover problem on graphs with maximum degree 4 and minimum degree at least 2, parameterized by $r = k-n/3$. We give an algorithm for this problem whose running time is $O^*(1.6253^r)$. As a corollary, we obtain an $O^*(1.2403^k)$-time algorithm for vertex cover on graphs with maximum degree 4.