Sparse induced subgraphs in P_6-free graphs

TL;DR

This paper demonstrates that several computational problems involving finding large sparse induced subgraphs with properties definable in CMSO2 are solvable in polynomial time within P_6-free graphs, extending previous results.

Contribution

The authors introduce a new framework based on listing carvers instead of potential maximal cliques, enabling polynomial-time solutions for these problems in P_6-free graphs.

Findings

Polynomial-time algorithms for Feedback Vertex Set in P_6-free graphs

Generalization of potential maximal cliques framework

Extension of previous results from P_5-free to P_6-free graphs

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.