We prove that a number of computational problems that ask for the largest sparse induced subgraph satisfying some property definable in CMSO2 logic, most notably Feedback Vertex Set, are polynomial-time solvable in the class of $P_6$-free graphs. This generalizes the work of Grzesik, Klimo\v{s}ov\'{a}, Pilipczuk, and Pilipczuk on the Maximum Weight Independent Set problem in $P_6$-free graphs~[SODA 2019, TALG 2022], and of Abrishami, Chudnovsky, Pilipczuk, Rz\k{a}\.zewski, and Seymour on problems in $P_5$-free graphs~[SODA~2021]. The key step is a new generalization of the framework of potential maximal cliques. We show that instead of listing a large family of potential maximal cliques, it is sufficient to only list their carvers: vertex sets that contain the same vertices from the sought solution and have similar separation properties.

翻译：我们证明了一系列询问满足CMSO2逻辑可定义性质的最大稀疏导出子图的计算问题，特别是反馈顶点集问题，在$P_6$-自由图类中是多项式时间可解的。这推广了Grzesik、Klimošová、Pilipczuk和Pilipczuk关于$P_6$-自由图中最大权独立集问题的工作~[SODA 2019, TALG 2022]，以及Abrishami、Chudnovsky、Pilipczuk、Rzążewski和Seymour关于$P_5$-自由图中问题的工作~[SODA~2021]。关键步骤是对潜在极大团框架的新推广。我们证明，无需列出大量潜在极大团，仅需列出它们的"雕琢者"（即包含与所求解相同顶点且具有相似分离性质的顶点集）便足矣。