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

Maria Chudnovsky, Jadwiga Czyżewska, Kacper Kluk, Marcin Pilipczuk, Paweł Rzążewski

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 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.

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

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 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 -free graphs by Chudnovsky, McCarty, Pilipczuk, Pilipczuk, and Rzążewski [SODA 2024]. In this work we extend polynomial-time tractability of all such problems to -free graphs of bounded clique number.

Paper Structure

This paper contains 29 sections, 38 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Potential maximal cliques.
  3. MWIS in graphs excluding a long induced path.
  4. Our contribution.
  5. Technical overview
  6. Preliminaries
  7. PMCs and chordal completions
  8. Treedepth structures and treewidth
  9. MWIS
  10. Structural lemmas
  11. $\chi$-boundness
  12. 2VC-dimension
  13. Approximating minimal separators
  14. $P_7$-free bipartite graphs
  15. Completing to a chordal bipartite graph
  16. ...and 14 more sections

Key Result

Theorem 1.1

For every fixed integers $d,k$, and a CMSO$_2$ formula $\psi$, the $(\mathrm{tw} \leq d, \psi)$-MWIS problem can be solved in polynomial time in $P_7$-free graphs with clique number at most $k$.

Theorems & Definitions (89)

  • Theorem 1.1
  • Lemma 1.2
  • Theorem 1.3
  • Lemma 2.1: see Todinca
  • Theorem 2.2: see BouchitteT02Todinca
  • Lemma 2.3: see Lemma 2.12 in DBLP:conf/soda/ChudnovskyMPPR24
  • Lemma 2.4: see Lemma 2.13 in DBLP:conf/soda/ChudnovskyMPPR24
  • Lemma 2.5
  • proof
  • Lemma 2.6: see Lemma 4.2 in grzesik2020polynomialtime
  • ...and 79 more