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arXiv · 2307.07330

Sparse induced subgraphs in P_6-free graphs

Maria Chudnovsky·Rose McCarty·Marcin Pilipczuk·Michał Pilipczuk·Paweł Rzążewski
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Abstract

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 $P_6$-free graphs. This generalizes the work of Grzesik, Klimo\v{s}ov\'{a}, Pilipczuk, and Pilipczuk on the Maximum Weight Independent Set problem in $P_6$-free graphs~[SODA 2019, TALG 2022], and of Abrishami, Chudnovsky, Pilipczuk, Rz\k{a}\.zewski, and Seymour on problems in $P_5$-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.

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Maria Chudnovsky, Rose McCarty, Marcin Pilipczuk, Michał Pilipczuk, Paweł Rzążewski. 2023-07-14. Sparse induced subgraphs in P_6-free graphs. https://arxiv.org/abs/2307.07330

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