Recognizing $\mathbf{W_2}$ Graphs

Let $G$ be a graph. A set $S \subseteq V(G)$ is independent if its elements are pairwise non-adjacent. A vertex $v \in V(G)$ is shedding if for every independent set $S \subseteq V(G) \setminus N[v]$ there exists $u \in N(v)$ such that $S \cup \{u\}$ 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 $\alpha(G)$. A graph $G$ is well-covered if all its maximal independent sets are maximum, i.e. the size of every maximal independent set is $\alpha(G)$. The graph $G$ belongs to class $\mathbf{W_2}$ if every two pairwise disjoint independent sets in $G$ are included in two pairwise disjoint maximum independent sets. If a graph belongs to the class $\mathbf{W_2}$ then it is well-covered. Finding a maximum independent set in an input graph is an NP-complete problem. Recognizing well-covered graphs is co-NP-complete. The complexity status of deciding whether an input graph belongs to the $\mathbf{W_2}$ class is not known. Even when the input is restricted to well-covered graphs, the complexity status of recognizing graphs in $\mathbf{W_2}$ is not known. In this article, we investigate the connection between shedding vertices and $\mathbf{W_2}$ 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 $\mathbf{W_2}$ graphs.