On regular-stable graphs

Abstract

We introduce graphs G, with at least one maximum independent set of vertices, I, such that for all v in V(G)\I, the number of vertices in NG(v)∩I is constant. When this number of vertices is equal to λ we say that I has the λ-property and that G is λ-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 λ-property, herein called well-covered λ-regular-stable graphs.