Sparse induced subgraphs in $P_7$-free graphs of bounded clique number
For researchers in algorithmic graph theory, this provides a polynomial-time algorithm for a broader class of graphs, though it is an incremental extension of existing techniques.
The paper extends polynomial-time tractability of finding the largest sparse induced subgraph satisfying a CMSO₂-definable property to \(P_7\)-free graphs with bounded clique number, building on prior quasipolynomial and polynomial results for smaller paths.
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 Rzążewski [STOC 2021], and a recent polynomial-time algorithm for $P_6$-free graphs by Chudnovsky, McCarty, Pilipczuk, Pilipczuk, and Rzążewski [SODA 2024]. In this work we extend polynomial-time tractability of all such problems to $P_7$-free graphs of bounded clique number.