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 P6-free graphs. This generalizes the work of Grzesik, Klimo\v{s}ov\'{a}, Pilipczuk, and Pilipczuk on the Maximum Weight Independent Set problem in P6-free graphs~[SODA 2019, TALG 2022], and of Abrishami, Chudnovsky, Pilipczuk, Rz\k{a}\.zewski, and Seymour on problems in P5-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.
@article{arxiv.2307.07330,
title = {Sparse induced subgraphs in P_6-free graphs},
author = {Maria Chudnovsky and Rose McCarty and Marcin Pilipczuk and Michał Pilipczuk and Paweł Rzążewski},
journal= {arXiv preprint arXiv:2307.07330},
year = {2023}
}