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 Hartnell (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.

翻译：设 $k \geq 1$。若对于图 $G$ 中任意 $k$ 个两两不相交的独立顶点子集 $A_1, \dots, A_k$，均存在 $k$ 个两两不相交的最大独立集 $S_1, \dots, S_k$ 使得 $A_i \subseteq S_i$（$i \in [k]$），则称图 $G$ 为 $\mathbf{W_k}$ 图。Chvátal 与 Hartnell（1993年）以及 Sankaranarayana 与 Stewart（1992年）分别独立证明 $\mathbf{W_1}$ 图的识别问题是 co-NP-难的。本文推广该结果并回应 Levit 与 Tankus 近期提出的问题，证明当 $k \geq 2$ 时，$\mathbf{W_k}$ 图的识别问题亦是 co-NP-难的。另一方面，我们证明对于每个 $k \geq 2$，$\mathbf{W_k}$ 图的识别问题在团宽和树宽参数化意义下均为 FPT（固定参数可解）。最后，我们构造了非 $\mathbf{W_2}$ 图 $G$，使得对于 $G$ 中任意顶点 $v$ 及 $G - N[v]$ 中任意极大独立集 $S$，$N(v) \setminus S$ 中的最大独立集均由单顶点构成，从而反驳了 Levit 与 Tankus 的一个猜想。