  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.
