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，例如最大权独立集、反馈顶点集或顶点平面化，可以统一在寻找满足CMSO$_2$逻辑可定义性质的极大稀疏诱导子图的框架下。据信，所有能用此形式化表达的问题，在排除固定路径作为诱导子图的图中，均可在多项式时间内求解。这一信念得到了Gartland、Lokshtanov、Pilipczuk、Pilipczuk和Rz\k{a}\.zewski [STOC 2021]提出的拟多项式时间算法，以及Chudnovsky、McCarty、Pilipczuk、Pilipczuk和Rz\k{a}\.zewski [SODA 2024]最近针对$P_6$自由图提出的多项式时间算法的支持。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至具有有界团数的$P_7$自由图。