Recognizing $$\mathbf {W_2}$$ Graphs

Abstract

AbstractLet G be a graph. A set $$S \subseteq V(G)$$ S ⊆ V ( G ) is independent if its elements are pairwise nonadjacent. A vertex $$v \in V(G)$$ v ∈ V ( G ) is shedding if for every independent set $$S \subseteq V(G) {\setminus } N[v]$$ S ⊆ V ( G ) \ N [ v ] there exists $$u \in N(v)$$ u ∈ N ( v ) such that $$S \cup \{u\}$$ 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 $$\alpha (G)$$ 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 $$\alpha (G)$$ α ( G ) . The graph G belongs to class $$\mathbf {W_2}$$ W 2 if every two disjoint independent sets in G are included in two disjoint maximum independent sets. If a graph belongs to the class $$\mathbf {W_2}$$ W 2 , 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 $$\mathbf {W_2}$$ W 2 is co-NP-complete. However, when the input is restricted to well-covered graphs, the complexity status of recognizing graphs in $$\mathbf {W_2}$$ W 2 is still not known. In this article, we investigate the connection between shedding vertices and $$\mathbf {W_2}$$ W 2 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 $$\mathbf {W_2}$$ W 2 graphs.