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Polynomial bounds for pathwidth

Sepehr Hajebi

TL;DR

The paper resolves the pathwidth analogue of a prominent treewidth conjecture for hereditary graph classes by showing that if pathwidth is bounded by a function of the clique number, then it is in fact bounded by a polynomial in the clique number. The authors achieve this through a three-pronged approach: a pw-to-tw bound under forbidding induced binary trees (B_r), a polynomial block/separability framework, and a polynomial separability scheme built via a seedling method and Ramsey-type arguments. Central to the argument are technical developments relating pathwidth, treewidth, induced minor models, and separability (including new notions like (kappa, lambda)-blocks and zeta-jagged graphs), plus a meticulous combination with known grid-minor and planar graph results. The outcomes include algorithmic consequences such as quasi-polynomial-time solvability for MWIS in hereditary pw-omega-bounded classes, along with a set of equivalences and corollaries connecting to bipartite obstructions and 2-degenerate graphs. The work thus establishes a robust, polynomially-structured relationship between pathwidth and clique number in a broad class of hereditary graphs, contrasting sharply with the treewidth landscape and enabling practical implications for wide-ranging graph problems.

Abstract

Dallard, Milanič, and Štorgel conjectured that for a hereditary graph class $\mathcal{G}$, if there is some function $f:\mathbb{N}\to\mathbb{N}$ such that every graph $G\in \mathcal{G}$ with clique number $ω(G)$ has treewidth at most $f(ω(G))$, then there is a polynomial function $f$ with the same property. Chudnovsky and Trotignon refuted this conjecture in a strong sense, showing that neither polynomial nor any prescribed growth can be guaranteed in general. Here we prove that, in stark contrast, the analog of the Dallard-Milanič-Štorgel conjecture for pathwidth is true: For every hereditary graph class $\mathcal{G}$, if the pathwidth of every graph in $\mathcal{G}$ is bounded by some function of its clique number, then the pathwidth of every graph in $\mathcal{G}$ is bounded by a polynomial function of its clique number.

Polynomial bounds for pathwidth

TL;DR

The paper resolves the pathwidth analogue of a prominent treewidth conjecture for hereditary graph classes by showing that if pathwidth is bounded by a function of the clique number, then it is in fact bounded by a polynomial in the clique number. The authors achieve this through a three-pronged approach: a pw-to-tw bound under forbidding induced binary trees (B_r), a polynomial block/separability framework, and a polynomial separability scheme built via a seedling method and Ramsey-type arguments. Central to the argument are technical developments relating pathwidth, treewidth, induced minor models, and separability (including new notions like (kappa, lambda)-blocks and zeta-jagged graphs), plus a meticulous combination with known grid-minor and planar graph results. The outcomes include algorithmic consequences such as quasi-polynomial-time solvability for MWIS in hereditary pw-omega-bounded classes, along with a set of equivalences and corollaries connecting to bipartite obstructions and 2-degenerate graphs. The work thus establishes a robust, polynomially-structured relationship between pathwidth and clique number in a broad class of hereditary graphs, contrasting sharply with the treewidth landscape and enabling practical implications for wide-ranging graph problems.

Abstract

Dallard, Milanič, and Štorgel conjectured that for a hereditary graph class , if there is some function such that every graph with clique number has treewidth at most , then there is a polynomial function with the same property. Chudnovsky and Trotignon refuted this conjecture in a strong sense, showing that neither polynomial nor any prescribed growth can be guaranteed in general. Here we prove that, in stark contrast, the analog of the Dallard-Milanič-Štorgel conjecture for pathwidth is true: For every hereditary graph class , if the pathwidth of every graph in is bounded by some function of its clique number, then the pathwidth of every graph in is bounded by a polynomial function of its clique number.
Paper Structure (13 sections, 47 theorems, 41 equations, 1 figure)

This paper contains 13 sections, 47 theorems, 41 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background and the (weaker) main result.
  3. The stronger result and its consequences.
  4. Preliminaries
  5. Getting the good old stuff out of the way
  6. Pathwidth versus treewidth
  7. Treewidth versus separability
  8. Monochromatic components of bounded diameter
  9. Induced minor models of bipartite graphs
  10. Excluding subdivided complete graphs as subgraphs
  11. Proof of \ref{['thm:mainpolyblock']}
  12. Polynomial Separability
  13. Acknowledgments

Key Result

Theorem 1.1

Let $\mathcal{G}$ be a minor-closed class. Then every graph in $\mathcal{G}$ has bounded pathwidth if and only if $\mathcal{G}$ excludes a forest.

Figures (1)

  • Figure 1: Left: A subdivision of $\mathop{\mathrm{\text{\sf{B}}}}\nolimits_3$ (top) and its line graph (bottom). Middle: The $4$-wall (top) and the $4$-grid (bottom). Right: An induced model of the $3$-grid in the $5$-wall.

Theorems & Definitions (71)

  • Theorem 1.1: Robertson and Seymour GMI
  • Theorem 1.2: Robertson and Seymour GMV
  • Conjecture 1.3: Dallard, Milanič, Štorgel DMS
  • Theorem 1.4: Chudnovsky and Trotignon; Theorem 5.1 in CT
  • Theorem 1.5
  • Theorem 1.5
  • Conjecture 1.6: Hajebi chordalehf; Disproved by Chudnovsky and Trotignon CT
  • Conjecture 1.7: Cocks; Equivalent to Conjecture 1.5 in cocks
  • Corollary 1.8
  • proof
  • ...and 61 more