Critical Graphs for Minimum Vertex Cover

TL;DR

This paper introduces the concept of critical graphs for the Minimum Vertex Cover problem, providing a new graph-generation method, systematically finding small critical graphs, and demonstrating their use in benchmarking heuristics.

Contribution

It extends the concept of critical graphs to Minimum Vertex Cover, offering a parametrized generation process and empirical evaluation for benchmarking algorithms.

Findings

Parametrized graph-generation preserves known minimum covers.

Systematic search identified small critical graphs.

Critical graphs are effective for benchmarking heuristics.

Abstract

In the context of the chromatic-number problem, a critical graph is an instance where the deletion of any element would decrease the graph's chromatic number. Such instances have shown to be interesting objects of study for deepen the understanding of the optimization problem. This work introduces critical graphs in context of Minimum Vertex Cover. We demonstrate their potential for the generation of larger graphs with hidden a priori known solutions. Firstly, we propose a parametrized graph-generation process which preserves the knowledge of the minimum cover. Secondly, we conduct a systematic search for small critical graphs. Thirdly, we illustrate the applicability for benchmarking purposes by reporting on a series of experiments using the state-of-the-art heuristic solver NuMVC.