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Tree Independence Number IV. Even-hole-free Graphs

Maria Chudnovsky, Peter Gartland, Sepehr Hajebi, Daniel Lokshtanov, Sophie Spirkl

TL;DR

This work proves that every $n$-vertex even-hole-free graph has tree independence number bounded by $c\log^{10} n$ for some constant $c>0$, implying that a tree decomposition with bags of independence number at most $c\log^{10} n$ exists. The authors develop a suite of structural tools—pyramids, wheels, loaded pyramids, star cutsets, tree-strip systems, and a central-bag method—to derive polylogarithmic clique-separator bounds that translate into balanced separators. By combining local-to-global separation results with a hub-based decomposition and a central-bag framework, they obtain a quasi-polynomial-time algorithmic regime for Maximum Weight Independent Set and several related problems on even-hole-free graphs, tightening the link between structural graph theory and algorithmic tractability. The approach builds on and extends prior work on logarithmic treewidth for $K_t$-free even-hole-free graphs, generalizing the methodology to the independence-number width and CMSO$_2$-based algorithmic consequences. Overall, the paper advances our understanding of how forbidding even holes constrains graph structure to yield powerful algorithmic consequences in quasi-polynomial time.

Abstract

We prove that the tree independence number of every even-hole-free graph is at most polylogarithmic in its number of vertices. More explicitly, we prove that there exists a constant c>0 such that for every integer n>1 every n-vertex even-hole-free graph has a tree decomposition where each bag has stability (independence) number at most c log^10 n. This implies that the Maximum Weight Independent Set problem, as well as several other natural algorithmic problems that are known to be NP-hard in general, can be solved in quasi-polynomial time if the input graph is even-hole-free.

Tree Independence Number IV. Even-hole-free Graphs

TL;DR

This work proves that every -vertex even-hole-free graph has tree independence number bounded by for some constant , implying that a tree decomposition with bags of independence number at most exists. The authors develop a suite of structural tools—pyramids, wheels, loaded pyramids, star cutsets, tree-strip systems, and a central-bag method—to derive polylogarithmic clique-separator bounds that translate into balanced separators. By combining local-to-global separation results with a hub-based decomposition and a central-bag framework, they obtain a quasi-polynomial-time algorithmic regime for Maximum Weight Independent Set and several related problems on even-hole-free graphs, tightening the link between structural graph theory and algorithmic tractability. The approach builds on and extends prior work on logarithmic treewidth for -free even-hole-free graphs, generalizing the methodology to the independence-number width and CMSO-based algorithmic consequences. Overall, the paper advances our understanding of how forbidding even holes constrains graph structure to yield powerful algorithmic consequences in quasi-polynomial time.

Abstract

We prove that the tree independence number of every even-hole-free graph is at most polylogarithmic in its number of vertices. More explicitly, we prove that there exists a constant c>0 such that for every integer n>1 every n-vertex even-hole-free graph has a tree decomposition where each bag has stability (independence) number at most c log^10 n. This implies that the Maximum Weight Independent Set problem, as well as several other natural algorithmic problems that are known to be NP-hard in general, can be solved in quasi-polynomial time if the input graph is even-hole-free.
Paper Structure (17 sections, 44 theorems, 18 equations, 1 figure)

This paper contains 17 sections, 44 theorems, 18 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof Outline
  4. Separating Two Vertices
  5. Organization of the Proof
  6. Jumps on pyramids
  7. Star cutsets from wheels
  8. Star cutsets from loaded pyramids
  9. Tree strip systems
  10. From local to global separators
  11. Handling dangerous triples
  12. Separating a pair of vertices: the hub-free case
  13. Stable sets of safe hubs
  14. Bounding the number of non-hubs
  15. Separating a pair of vertices: a bound using clique size
  16. ...and 2 more sections

Key Result

Theorem 1.1

There exists a constant $c$ such that for every integer $n>1$ every $n$-vertex even-hole-free graph has tree independence number at most $c \log^{10} n$.

Figures (1)

  • Figure 1: Proof of Lemma \ref{['laminar']}.

Theorems & Definitions (67)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 3.1
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Theorem 4.3
  • proof
  • ...and 57 more