Sparse induced subgraphs in $P_7$-free graphs of bounded clique number

TL;DR

This paper extends polynomial-time algorithms for finding large sparse induced subgraphs satisfying CMSO$_2$ properties to $P_7$-free graphs with bounded clique number, broadening the class of graphs where these problems are efficiently solvable.

Contribution

It introduces a polynomial-time algorithm for solving CMSO$_2$ definable problems on $P_7$-free graphs with bounded clique number, expanding previous results from $P_6$-free graphs.

Findings

Polynomial-time algorithm for $P_7$-free graphs with bounded clique number.

Extension of tractability results from $P_6$-free to $P_7$-free graphs.

Supports a wide range of problems including Max Weight Independent Set and Feedback Vertex Set.

Abstract

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.