On Regular-Stable Graphs

Abstract

We introduce graphs \(G\) with at least one maximum independent set of vertices \(I\), such that \(\forall v \in V(G) \setminus I\), the number of vertices in \(N_G(v) \cap I\) is constant. When this number of vertices is equal to \(\lambda\), we say that \(I\) has the \(\lambda\)-property and that \(G\) is \(\lambda\)-regular-stable. Furthermore, we extend the study of this property to the well-covered graphs (that is, graphs where all maximal independent sets of vertices have the same cardinality). In this study, we consider well-covered graphs for which all maximal independent sets of vertices have the \(\lambda\)-property, herein called well-covered \(\lambda\)-regular-stable graphs.