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Graph Parameters, Universal Obstructions, and WQO

Christophe Paul, Evangelos Protopapas, Dimitrios M. Thilikos

TL;DR

The parametric framework for obtaining obstruction characterizations of graph parameters with respect to a quasi-ordering is established and combinatorial objects that capture the approximate behaviour of graph parameters are defined.

Abstract

We establish a parametric framework for obtaining obstruction characterizations of graph parameters with respect to a quasi-ordering $\leqslant$ on graphs. At the center of this framework lies the concept of a $\leqslant$-parametric graph: a non $\leqslant$-decreasing sequence $\mathscr{G} = \langle \mathscr{G}_{t} \rangle_{t \in \mathbb{N}}$ of graphs indexed by non-negative integers. Parametric graphs allow us to define combinatorial objects that capture the approximate behaviour of graph parameters. A finite set $\mathfrak{G}$ of $\leqslant$-parametric graphs is a $\leqslant$-universal obstruction for a parameter $\mathsf{p}$ if there exists a function $f \colon \mathbb{N} \to \mathbb{N}$ such that, for every $k \in \mathbb{N}$ and every graph $G$, 1) if $\mathsf{p}(G) \leq k$, then for every $\mathscr{G} \in \mathfrak{G},$ $\mathscr{G}_{f(k)} \not\leqslant G$, and 2) if for every $\mathscr{G} \in \mathfrak{G},$ $\mathscr{G}_{k} \not\leqslant G$, then $\mathsf{p}(G) \leq f(k).$ To solidify our point of view, we identify sufficient order-theoretic conditions that guarantee the existence of universal obstructions and in this case we examine algorithmic implications on the existence of fixed-parameter tractable algorithms. Our parametric framework has further implications related to finite obstruction characterizations of properties of graph classes. A $\leqslant$-class property is defined as any set of $\leqslant$-closed graph classes that is closed under set inclusion. By combining our parametric framework with established results from order theory, we derive a precise order-theoretic characterization that ensures $\leqslant$-class properties can be described in terms of the exclusion of a finite set of $\leqslant$-parametric graphs.

Graph Parameters, Universal Obstructions, and WQO

TL;DR

The parametric framework for obtaining obstruction characterizations of graph parameters with respect to a quasi-ordering is established and combinatorial objects that capture the approximate behaviour of graph parameters are defined.

Abstract

We establish a parametric framework for obtaining obstruction characterizations of graph parameters with respect to a quasi-ordering on graphs. At the center of this framework lies the concept of a -parametric graph: a non -decreasing sequence of graphs indexed by non-negative integers. Parametric graphs allow us to define combinatorial objects that capture the approximate behaviour of graph parameters. A finite set of -parametric graphs is a -universal obstruction for a parameter if there exists a function such that, for every and every graph , 1) if , then for every , and 2) if for every , then To solidify our point of view, we identify sufficient order-theoretic conditions that guarantee the existence of universal obstructions and in this case we examine algorithmic implications on the existence of fixed-parameter tractable algorithms. Our parametric framework has further implications related to finite obstruction characterizations of properties of graph classes. A -class property is defined as any set of -closed graph classes that is closed under set inclusion. By combining our parametric framework with established results from order theory, we derive a precise order-theoretic characterization that ensures -class properties can be described in terms of the exclusion of a finite set of -parametric graphs.
Paper Structure (68 sections, 61 theorems, 38 equations, 6 figures)

This paper contains 68 sections, 61 theorems, 38 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Well-quasi-ordering and graph algorithms.
  3. Universal obstructions for graph parameters.
  4. Class properties and graph parameters.
  5. Algorithmic consequences of the parametric viewpoint.
  6. Parameterized algorithms.
  7. Examples.
  8. Background concepts
  9. Order theory
  10. Binary relations.
  11. Quasi-orderings, partial-orderings, and equivalence relations.
  12. Well-founded quasi-orderings.
  13. Well-quasi-orderings.
  14. Dilworth's theorem.
  15. Closed sets.
  16. ...and 53 more sections

Key Result

Theorem 2.1

Let $(X, \leqslant)$ be a partial-ordering. Then $(X, \leqslant)$ has finite width $w$ if and only if it can be partitioned into $w$ many $\leqslant$-chains.

Figures (6)

  • Figure 1: An illustration of the Rado structure.
  • Figure 2: The minor-parametric $\mathscr{T}=\langle \mathscr{T}_1,\mathscr{T}_2,\mathscr{T}_3,\mathscr{T}_4,\ldots\rangle$ of complete ternary trees.
  • Figure 3: The minor-parametric graph $\mathscr{T}^a = \langle \mathscr{T}^a_2,\mathscr{T}^a_3,\mathscr{T}^a_4,\mathscr{T}^a_5,\ldots\rangle.$
  • Figure 4: The minor-parametric graph $\mathscr{T}^{a*}=\langle \mathscr{T}^{a*}_2,\mathscr{T}^{a*}_3,\mathscr{T}^{a*}_4,\mathscr{T}^{a*}_5,\ldots\rangle.$
  • Figure 5: The graphs in the two minor-obstruction sets in $\hbox{\rm pobs}_{\leqslant_{\mathsf{m}}}({\sf bi\hbox{-}pw}\xspace).$
  • ...and 1 more figures

Theorems & Definitions (113)

  • Theorem 2.1: Dilworth's Theorem
  • Proposition 1: SchmitzS12Algorithmic
  • Proposition 2: SchmitzS12Algorithmic
  • Proposition 3: Lemma 5.3, KabilP92Une
  • Proposition 4: SchmitzS12Algorithmic
  • Proposition 5: SchmitzS12Algorithmic
  • Definition 1: Smyth extension
  • Proposition 6: Marcone01Fine
  • Definition 2: Jancar99ANote
  • Theorem 2.2: Jancar99ANote
  • ...and 103 more