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An Overview of Universal Obstructions for Graph Parameters

Christophe Paul, Evangelos Protopapas, Dimitrios M. Thilikos

TL;DR

The paper presents a unifying framework of universal obstructions to classify graph parameters that are monotone under various quasi-orders. It formalizes class obstructions, parametric obstructions, and universal obstructions, and then surveys extensive results across minor, immersion, and vertex-minor monotone parameters, illustrating how these obstructions yield equivalent or near-equivalent parameter descriptions. It provides concrete universal obstructions for core parameters (treewidth, pathwidth, treedepth, rankwidth, etc.) and discusses how these obstructions inform hierarchy relations and potential polynomial gaps, as well as the limitations in non-wqo settings. The work highlights the potential algorithmic benefits of this framework, including automatic derivation of fixed-parameter approximations and unified comparisons across parameters, and outlines key open questions about finiteness of obstructions and the reach of universal obstructions in broader settings.

Abstract

In a recent work, we introduced a parametric framework for obtaining obstruction characterizations of graph parameters with respect to a quasi-ordering $\leqslant$ on graphs. Towards this, we proposed the concepts of class obstruction, parametric obstruction, and universal obstruction as combinatorial objects that determine the approximate behaviour of a graph parameter. In this work, we explore its potential as a unifying framework for classifying graph parameters. Under this framework, we survey existing graph-theoretic results on many known graph parameters. Additionally, we provide some unifying results on their classification.

An Overview of Universal Obstructions for Graph Parameters

TL;DR

The paper presents a unifying framework of universal obstructions to classify graph parameters that are monotone under various quasi-orders. It formalizes class obstructions, parametric obstructions, and universal obstructions, and then surveys extensive results across minor, immersion, and vertex-minor monotone parameters, illustrating how these obstructions yield equivalent or near-equivalent parameter descriptions. It provides concrete universal obstructions for core parameters (treewidth, pathwidth, treedepth, rankwidth, etc.) and discusses how these obstructions inform hierarchy relations and potential polynomial gaps, as well as the limitations in non-wqo settings. The work highlights the potential algorithmic benefits of this framework, including automatic derivation of fixed-parameter approximations and unified comparisons across parameters, and outlines key open questions about finiteness of obstructions and the reach of universal obstructions in broader settings.

Abstract

In a recent work, we introduced a parametric framework for obtaining obstruction characterizations of graph parameters with respect to a quasi-ordering on graphs. Towards this, we proposed the concepts of class obstruction, parametric obstruction, and universal obstruction as combinatorial objects that determine the approximate behaviour of a graph parameter. In this work, we explore its potential as a unifying framework for classifying graph parameters. Under this framework, we survey existing graph-theoretic results on many known graph parameters. Additionally, we provide some unifying results on their classification.
Paper Structure (76 sections, 45 theorems, 82 equations, 39 figures)

This paper contains 76 sections, 45 theorems, 82 equations, 39 figures.

Table of Contents

  1. Introduction
  2. Importance of a parameter.
  3. Comparing graph parameters.
  4. Parameter description.
  5. Capturing the asymptotic behavior of monotone parameters.
  6. Universal obstructions: basic definitions and results
  7. General concepts
  8. Graph parameters
  9. Class and parametric obstructions.
  10. Parametric graphs
  11. Comparing parameters using Smyth extensions
  12. Basic definitions
  13. Obstructions of minor-monotone parameters
  14. Warm up
  15. Number of vertices.
  16. ...and 61 more sections

Key Result

Proposition 1

If a quasi-ordering $\leqslant$ is a well-quasi-ordering on $\mathcal{G}_{{\text{\rm all}}}$, then for every $\leqslant$-monotone parameter, the set $\hbox{\rm cobs}_\leqslant({\sf p})$ (and therefore also $\hbox{\rm pobs}_\leqslant({\sf p})$) exists.

Figures (39)

  • Figure 1: The minor-parametric graph $\mathscr{A}=\langle \mathscr{A}_2, \mathscr{A}_3, \mathscr{A}_4, \mathscr{A}_5, \mathscr{A}_6 \ldots \rangle$ of annulus grids.
  • Figure 2: The minor-parametric graph $\mathscr{T} = \langle \mathscr{T}_1, \mathscr{T}_2, \mathscr{T}_3, \mathscr{T}_4, \ldots \rangle$ of complete ternary trees.
  • Figure 3: The minor-parametric graph $\mathscr{P} = \langle \mathscr{P}_1, \mathscr{P}_2, \mathscr{P}_3, \mathscr{P}_4, \mathscr{P}_5, \ldots \rangle$ of paths.
  • Figure 4: The minor-parametric graph $\mathscr{K} = \langle \mathscr{K}_2, \mathscr{K}_3, \mathscr{K}_4, \mathscr{K}_5, \mathscr{K}_6 \ldots \rangle$ of clique grids.
  • Figure 5: The minor-parametric graph $\mathscr{S} = \langle \mathscr{S}_2, \mathscr{S}_3, \mathscr{S}_4, \mathscr{S}_5, \mathscr{S}_6 \ldots \rangle$ of singly-crossing grids.
  • ...and 34 more figures

Theorems & Definitions (59)

  • Definition 1: Class obstruction
  • Definition 2: Parametric obstruction
  • Proposition 1
  • Definition 3
  • Definition 4: Universal obstruction
  • Proposition 2: PaulPT2023GraphParameters
  • Proposition 3: PaulPT2023GraphParameters
  • Proposition 4
  • proof
  • Proposition 5
  • ...and 49 more