Improved Complexity Bound of Vertex Cover for Low degree Graph

TL;DR

This paper introduces a novel reduction technique called 'real-cycle' subset to improve the parameterized complexity bounds for finding minimum vertex covers in low-degree graphs, achieving the best known bounds to date.

Contribution

The paper presents a new 'real-cycle' reduction method that, combined with existing techniques, significantly improves the complexity bounds for vertex cover in degree-3 graphs.

Findings

Achieved a complexity bound of O(1.15855^k) using 'real-cycle' reductions.

Further improved the bound to O(1.1504^k) with additional techniques.

Established the current best complexity bounds for this problem.

Abstract

In this paper, we use a new method to decrease the parameterized complexity bound for finding the minimum vertex cover of connected max-degree-3 undirected graphs. The key operation of this method is reduction of the size of a particular subset of edges which we introduce in this paper and is called as "real-cycle" subset. Using "real-cycle" reductions alone we compute a complexity bound $O(1.15855^k)$ where $k$ is size of the optimal vertex cover. Combined with other techniques, the complexity bound can be further improved to be $O(1.1504^k)$. This is currently the best complexity bound.