  Let $k \geq 1$. A graph $G$ is $\mathbf{W_k}$ if for any $k$ pairwise
disjoint independent vertex subsets $A_1, \dots, A_k$ in $G$, there exist $k$
pairwise disjoint maximum independent sets $S_1, \dots, S_k$ in $G$ such that
$A_i \subseteq S_i$ for $i \in [k]$. Recognizing $\mathbf{W_1}$ graphs is
co-NP-hard, as shown by Chv\'atal and Slater (1993) and, independently, by
Sankaranarayana and Stewart (1992). Extending this result and answering a
recent question of Levit and Tankus, we show that recognizing $\mathbf{W_k}$
graphs is co-NP-hard for $k \geq 2$. On the positive side, we show that
recognizing $\mathbf{W_k}$ graphs is, for each $k\geq 2$, FPT parameterized by
clique-width and by tree-width. Finally, we construct graphs $G$ that are not
$\mathbf{W_2}$ such that, for every vertex $v$ in $G$ and every maximal
independent set $S$ in $G - N[v]$, the largest independent set in $N(v)
\setminus S$ consists of a single vertex, thereby refuting a conjecture of
Levit and Tankus.
