W2-graphs and shedding vertices

Abstract

A graph is well-covered if all its maximal independent sets are of the same size (Plummer, 1970). A well-covered graph is 1-well-covered if the deletion of every vertex leaves a graph which is well-covered as well (Staples, 1975).A graph G belongs to class Wn if every n pairwise disjoint independent sets in G are included in n pairwise disjoint maximum independent sets (Staples, 1975). Clearly, W1 is the family of all well-covered graphs. It turns out that G∈W2 if and only if it is a 1-well-covered graph without isolated vertices.We show that deleting a shedding vertex does not change the maximum size of a maximal independent set including a given A∈Ind(G) in a graph G, where Ind(G) is the family of all the independent sets. Specifically, for well-covered graphs, it means that the vertex v is shedding if and only if G−v is well-covered.