Stable Well-Covered Graphs

Abstract

A graph \(G\) is said to be well-covered if every maximal independent set of \(G\) is of the same size. It has been shown that characterizing well-covered graphs is a co-NP-complete problem. In an effort to characterize some of these graphs, different subclasses of well-covered graphs have been studied. In this paper, we will introduce the subclass of stable well-covered graphs, which are well-covered graphs that remain well-covered with the addition of any edge. Some properties of stable well-covered graphs are given. In addition, the relationships between stable well-covered graphs and some other subclasses of well-covered graphs, including the surprising equivalence between stable well-covered graphs and other known subclasses, are proved.