On graphs admitting two disjoint maximum independent sets

TL;DR

This paper investigates conditions under which graphs admit two disjoint maximum independent sets, building on prior work that characterized such graphs and addressing the NP-completeness of related decision problems.

Contribution

It provides new conditions that guarantee the existence of two disjoint maximum independent sets in graphs, extending previous characterizations.

Findings

Characterization of graphs with two disjoint maximum independent sets

Conditions ensuring the existence of such sets

Insights into the computational complexity of related problems

Abstract

An independent set A is maximal if it is not a proper subset of an independent set, while A is maximum if it has a maximum size. The problem of whether a graph has a pair of disjoint maximal independent sets was introduced by C. Berge in early 70's. The class of graphs for which every induced subgraph admits two disjoint maximal independent sets was characterized in (Shaudt, 2015). It is known that deciding whether a graph has two disjoint maximal independent sets is a NP-complete problem (Henning et al., 2009). In this paper, we are focused on finding conditions ensuring the existence of two disjoint maximum independent sets.