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A finite presentation of graphs of treewidth at most three

Amina Doumane, Samuel Humeau, Damien Pous

TL;DR

The paper resolves the open question of a finite equational presentation for graphs of treewidth at most $3$ by introducing a syntax with interfaces and a finite axiom set. The approach relies on a two-tier structural analysis: a canonical decomposition into full prime components and a non-deterministic parsing phase, plus a careful handling of anchors and separation pairs to ensure all parses are equivalent under a finite set of equations. The main technical contribution is proving forget-point agreement for non-atomic full prime graphs, enabling completeness with a finite axiom system tailored to treewidth $3$. This advances understanding of graph syntaxes and could inform finite presentations for higher treewidths and connections to minor-closed graph families.

Abstract

We provide a finite equational presentation of graphs of treewidth at most three, solving an instanceof an open problem by Courcelle and Engelfriet. We use a syntax generalising series-parallel expressions, denoting graphs with a small interface. Weintroduce appropriate notions of connectivity for such graphs (components, cutvertices, separationpairs). We use those concepts to analyse the structure of graphs of treewidth at most three, showinghow they can be decomposed recursively, first canonically into connected parallel components, andthen non-deterministically. The main difficulty consists in showing that all non-deterministic choicescan be related using only finitely many equational axioms.

A finite presentation of graphs of treewidth at most three

TL;DR

The paper resolves the open question of a finite equational presentation for graphs of treewidth at most by introducing a syntax with interfaces and a finite axiom set. The approach relies on a two-tier structural analysis: a canonical decomposition into full prime components and a non-deterministic parsing phase, plus a careful handling of anchors and separation pairs to ensure all parses are equivalent under a finite set of equations. The main technical contribution is proving forget-point agreement for non-atomic full prime graphs, enabling completeness with a finite axiom system tailored to treewidth . This advances understanding of graph syntaxes and could inform finite presentations for higher treewidths and connections to minor-closed graph families.

Abstract

We provide a finite equational presentation of graphs of treewidth at most three, solving an instanceof an open problem by Courcelle and Engelfriet. We use a syntax generalising series-parallel expressions, denoting graphs with a small interface. Weintroduce appropriate notions of connectivity for such graphs (components, cutvertices, separationpairs). We use those concepts to analyse the structure of graphs of treewidth at most three, showinghow they can be decomposed recursively, first canonically into connected parallel components, andthen non-deterministically. The main difficulty consists in showing that all non-deterministic choicescan be related using only finitely many equational axioms.

Paper Structure

This paper contains 11 sections, 27 theorems, 13 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Graphs, treewidth
  3. Graph operations
  4. Terms and axioms
  5. Outline of the completeness proof
  6. Series decompositions
  7. Anchors
  8. Parsing on anchors at arity three
  9. Parsing on anchors at lower arities
  10. Separation pairs
  11. Conclusion and future work

Key Result

Proposition 3

For all graphs $G,H$ and all substitutions $\sigma,\rho$, if $G\simeq H$ and $\sigma(a)\simeq\rho(a)$ for all letters $a$, then $G\sigma\simeq H\rho$.

Figures (6)

  • Figure 1: Two parsings of a given graph.
  • Figure 2: Separation pairs and graphs with several separation pairs.
  • Figure 3: Forget axioms for anchors (swap and kite); $x,y$ are the inner vertices.
  • Figure 4: Forget axioms for hard graphs (cross and domino); $x,y$ are the topmost inner vertices.
  • Figure 5: The series operations ${\rm s}(;d)$, ${\rm s}(a;d)$, ${\rm s}(a,b;d)$, and ${\rm s}(a,b,c;d)$.
  • ...and 1 more figures

Theorems & Definitions (35)

  • Definition 1
  • Definition 2
  • Proposition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Lemma 8
  • Proposition 9
  • Definition 11: Finite axiomatisation
  • ...and 25 more