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Tree independence number II. Three-path-configurations

Maria Chudnovsky, Sepehr Hajebi, Daniel Lokshtanov, Sophie Spirkl

Abstract

A three-path-configuration is a graph consisting of three pairwise internally-disjoint paths the union of every two of which is an induced cycle of length at least four. A graph is 3PC-free if no induced subgraph of it is a three-path-configuration. We prove that 3PC-free graphs have poly-logarithmic tree-independence number. More explicitly, we show that there exists a constant $c$ such that every $n$-vertex 3PC-free graph graph has a tree decomposition in which every bag has stability number at most $c (\log n)^2$. 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 3PC-free.

Tree independence number II. Three-path-configurations

Abstract

A three-path-configuration is a graph consisting of three pairwise internally-disjoint paths the union of every two of which is an induced cycle of length at least four. A graph is 3PC-free if no induced subgraph of it is a three-path-configuration. We prove that 3PC-free graphs have poly-logarithmic tree-independence number. More explicitly, we show that there exists a constant such that every -vertex 3PC-free graph graph has a tree decomposition in which every bag has stability number at most . 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 3PC-free.
Paper Structure (10 sections, 23 theorems, 11 equations, 1 figure)

This paper contains 10 sections, 23 theorems, 11 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof outline and organization
  3. Structural results
  4. Dominated balanced separators
  5. Separating a pair of vertices
  6. Large stable subsets in neighborhoods
  7. From domination to stability
  8. The proof of Theorem \ref{['treealpha']}
  9. Algorithmic consequences
  10. Acknowledgments

Key Result

Theorem 1.1

For every integer $t>0$ there exists a constant $c(t)$ such that for every $n$-vertex graph $G \in \mathcal{C}$ that contains no clique of size $t$, $\mathop{\mathrm{tw}}\nolimits(G)\leq c(t) \log n$.

Figures (1)

  • Figure 1: The three-path-configurations. From left to right: A theta, a pyramid, a prism and a pinched prism (dashed lines depict paths of non-zero length).

Theorems & Definitions (35)

  • Theorem 1.1: Abrishami, Chudnovsky, Hajebi, Spirkl tw3
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • proof
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3: Chudnovsky, Gartland, Hajebi, Lokshtanov, Spirkl; Theorem 5.2 in tw15
  • Lemma 3.4
  • ...and 25 more