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 Rz\k{a}\.zewski [STOC 2021], and a recent polynomial-time algorithm for $P_6$-free graphs by Chudnovsky, McCarty, Pilipczuk, Pilipczuk, and Rz\k{a}\.zewski [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，例如最大权重独立集、反馈顶点集或顶点平面化问题，均可统一归约为寻找满足CMSO$_2$逻辑可定义性质的**最大稀疏诱导子图**这一框架。学界普遍认为，凡可通过此形式化方法表达的问题，均能在**排除固定路径作为诱导子图**的图中以多项式时间求解。这一信念得到了Gartland、Lokshtanov、Pilipczuk、Pilipczuk与Rz\k{a}żewski [STOC 2021]提出的拟多项式时间算法，以及Chudnovsky、McCarty、Pilipczuk、Pilipczuk与Rz\k{a}żewski [SODA 2024]近期针对$P_6$-自由图提出的多项式时间算法的支持。本工作中，我们将**所有此类问题的多项式时间可解性**扩展至**有界团数的$P_7$-自由图**。