Recognizing W2 Graphs

Let G be a graph. A set S⊆V(G) is independent if its elements are pairwise nonadjacent. A vertex v∈V(G) is shedding if for every independent set S⊆V(G)\N[v] there exists u∈N(v) such that S∪{u} is independent. An independent set S is maximal if it is not contained in another independent set. An independent set S is maximum if the size of every independent set of G is not bigger than |S|. The size of a maximum independent set of G is denoted α(G). A graph G is well-covered if all its maximal independent sets are maximum, i.e. the size of every maximal independent set is α(G). The graph G belongs to class W₂ if every two disjoint independent sets in G are included in two disjoint maximum independent sets. If a graph belongs to the class W₂, then it is well-covered. Finding a maximum independent set in an input graph is a well-known NP-hard problem. Recognizing well-covered graphs is co-NP-complete. Recently, it was proved that deciding whether an input graph belongs to the class W₂ is co-NP-complete. However, when the input is restricted to well-covered graphs, the complexity status of recognizing graphs in W₂ is still not known. In this article, we investigate the connection between shedding vertices and W₂ graphs. On the one hand, we prove that recognizing shedding vertices is co-NP-complete. On the other hand, we find polynomial solutions for restricted cases of the problem. We also supply polynomial characterizations of several families of W₂ graphs.