Many natural computational problems, including e.g. Max Weight Independent Set, Feedback Vertex Set, or Vertex Planarization, can be unified under an umbrella of finding the largest sparse induced subgraph, that satisfies some property definable in CMSO$_2$ logic. It is believed that each problem expressible with this formalism can be solved in polynomial time in graphs that exclude a fixed path as an induced subgraph. This belief is supported by the existence of a quasipolynomial-time algorithm by Gartland, Lokshtanov, Pilipczuk, Pilipczuk, and Rzążewski [STOC 2021], and a recent polynomial-time algorithm for $P_6$-free graphs by Chudnovsky, McCarty, Pilipczuk, Pilipczuk, and Rzążewski [SODA 2024]. In this work we extend polynomial-time tractability of all such problems to $P_7$-free graphs of bounded clique number.

翻译：许多自然计算问题，例如最大加权独立集、反馈顶点集或顶点平面化，都可以统一归为寻找最大的稀疏导出子图，且该子图满足 CMSO$_2$ 逻辑可定义的某些性质。人们普遍认为，每个可用该形式化表达的问题，在排除固定路径作为导出子图的图中都能在多项式时间内求解。这一观点得到了以下研究的支持：Gartland、Lokshtanov、Pilipczuk、Pilipczuk 和 Rzążewski [STOC 2021] 提出的拟多项式时间算法，以及 Chudnovsky、McCarty、Pilipczuk、Pilipczuk 和 Rzążewski [SODA 2024] 近期针对 $P_6$-free 图的多项式时间算法。在本工作中，我们将所有此类问题的多项式时间可解性扩展到具有有界团数的 $P_7$-free 图。