Graphs in which the Maxine heuristic produces a maximum independent set

TL;DR

This paper investigates the Maxine heuristic's effectiveness in producing maximum independent sets in graphs, providing a classification of graphs where the heuristic yields optimal results, thus advancing understanding of graph algorithms.

Contribution

It refines the classification of graphs for which the Maxine heuristic produces maximum independent sets, building on prior work by Barrus and Molnar.

Findings

Identifies conditions under which the Maxine heuristic yields maximum independent sets

Provides a forbidden subgraph classification for optimal heuristic performance

Enhances theoretical understanding of heuristic bounds in graph theory

Abstract

The residue of a graph is the number of zeros left after iteratively applying the Havel-Hakimi algorithm to its degree sequence. Favaron, Mah\'{e}o, and Sacl\'{e} showed that the residue is a lower bound on the independence number. The Maxine heuristic reduces a graph to an independent set of size $M$. It has been shown that given a graph $G$, $M$ is bounded between the independence number and the residue of a graph for any application of the Maxine heuristic. We improve upon a forbidden subgraph classification of graphs such that $M$ is equal to the independence number given by Barrus and Molnar in 2015.