We prove a number of results related to the computational complexity of recognizing well-covered graphs. Let $k$ and $s$ be positive integers and let $G$ be a graph. Then $G$ is said - $\mathbf{W_k}$ if for any $k$ pairwise disjoint independent vertex sets $A_1, \dots, A_k$ 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 $i \in [k]$. - $\mathbf{E_s}$ if every independent set in $G$ of size at most $s$ is contained in a maximum independent set in $G$. Chv\'atal and Slater (1993) and Sankaranarayana and Stewart (1992) famously showed that recognizing $\mathbf{W_1}$ graphs or, equivalently, well-covered graphs is coNP-complete. We extend this result by showing that recognizing $\mathbf{W_{k+1}}$ graphs in either $\mathbf{W_k}$ or $\mathbf{E_s}$ graphs is coNP-complete. This answers a question of Levit and Tankus (2023) and strengthens a theorem of Feghali and Marin (2024). We also show that recognizing $\mathbf{E_{s+1}}$ graphs is $\Theta_2^p$-complete even in $\mathbf{E_s}$ graphs, where $\Theta_2^p = \text{P}^{\text{NP}[\log]}$ is the class of problems solvable in polynomial time using a logarithmic number of calls to a SAT oracle. This strengthens a theorem of Berg\'e, Busson, Feghali and Watrigant (2023). We also obtain the complete picture of the complexity of recognizing chordal $\mathbf{W_k}$ and $\mathbf{E_s}$ graphs which, in particular, simplifies and generalizes a result of Dettlaff, Henning and Topp (2023).

翻译：我们证明了与识别良好覆盖图的计算复杂性相关的若干结果。设$k$和$s$为正整数，$G$为一图。则称$G$为： - $\mathbf{W_k}$图：若对于$G$中任意$k$个两两不相交的独立顶点集$A_1, \dots, A_k$，存在$k$个两两不相交的最大独立集$S_1, \dots, S_k$，使得对所有$i \in [k]$有$A_i \subseteq S_i$。 - $\mathbf{E_s}$图：若$G$中每个大小不超过$s$的独立集都包含于某个最大独立集中。 Chvátal与Slater（1993）以及Sankaranarayana与Stewart（1992）曾著名地证明，识别$\mathbf{W_1}$图（即等价于良好覆盖图）是coNP完全的。我们将此结果进行推广，证明在$\mathbf{W_k}$图或$\mathbf{E_s}$图中识别$\mathbf{W_{k+1}}$图是coNP完全的。这回答了Levit与Tankus（2023）的一个问题，并强化了Feghali与Marin（2024）的一个定理。我们还证明，即使在$\mathbf{E_s}$图中，识别$\mathbf{E_{s+1}}$图是$\Theta_2^p$完全的，其中$\Theta_2^p = \text{P}^{\text{NP}[\log]}$是通过对数次调用SAT预言机即可在多项式时间内求解的问题类。这强化了Bergé、Busson、Feghali与Watrigant（2023）的一个定理。此外，我们获得了识别弦图类中$\mathbf{W_k}$图与$\mathbf{E_s}$图复杂性的完整图像，这特别地简化并推广了Dettlaff、Henning与Topp（2023）的一个结果。