# Generalized domination structure in cubic graphs

## TL;DR

This paper introduces a novel polynomial-time method for determining the minimum dominating set in cubic graphs by leveraging structural properties and labeling schemes, improving upon the APX-complete complexity result.

## Contribution

It presents a new approach that solves the minimum dominating set problem in cubic graphs efficiently, which was previously known to be APX-complete.

## Key findings

- Polynomial-time algorithm for minimum dominating set in cubic graphs
- Structural characterization of dominating sets using labeling schemes
- Reduction of problem complexity from APX-complete to polynomial time

## Abstract

The minimum dominating set problem asks for a dominating set with minimum size. First, we determine some vertices contained in the minimum dominating set of a graph. By applying a particular scheme, we ensure that the resulting graph is 2-connected and the length of each formed induced cycle is 0 mod 3. We label every three vertices in the induced cycles of length 0 mod 3. Then there is a way of labeling in which the set of all labeled vertices is the minimum dominating set of the resulting graph, and is contained in the minimum dominating set of the original graph. We also consider the remaining vertices of the minimum dominating set of the original graph and determine all vertices contained in the minimum dominating set of a graph with maximum degree 3. The complexity of the minimum dominating set problem for cubic graphs was shown to be APX-complete in 2000 and this problem is solved by our arguments in polynomial time.