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 CMSO2 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 P6-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 P7-free graphs of bounded clique number.
@article{arxiv.2412.14836,
title = {Sparse induced subgraphs in $P_7$-free graphs of bounded clique number},
author = {Maria Chudnovsky and Jadwiga Czyżewska and Kacper Kluk and Marcin Pilipczuk and Paweł Rzążewski},
journal= {arXiv preprint arXiv:2412.14836},
year = {2026}
}