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.

翻译：设 $k \geq 1$。图 $G$ 称为 $\mathbf{W_k}$ 图，若对于 $G$ 中任意 $k$ 个两两不相交的独立顶点子集 $A_1, \dots, A_k$，存在 $G$ 中 $k$ 个两两不相交的最大独立集 $S_1, \dots, S_k$，使得对每个 $i \in [k]$ 均有 $A_i \subseteq S_i$。Chvátal 与 Slater（1993年）以及 Sankaranarayana 与 Stewart（1992年）独立证明了识别 $\mathbf{W_1}$ 图是 co-NP 难的。本文推广该结果并回应 Levit 与 Tankus 近期提出的问题，证明对于 $k \geq 2$，识别 $\mathbf{W_k}$ 图仍是 co-NP 难的。在积极方面，我们证明对于每个 $k \geq 2$，识别 $\mathbf{W_k}$ 图在团宽度和树宽参数化下是 FPT 的。最后，我们构造了非 $\mathbf{W_2}$ 的图 $G$，使得对 $G$ 中任意顶点 $v$ 及 $G - N[v]$ 中任意极大独立集 $S$，$N(v) \setminus S$ 中的最大独立集仅包含单个顶点，从而推翻了 Levit 与 Tankus 的一个猜想。