A finite presentation of graphs of treewidth at most three

TL;DR

This paper presents a finite algebraic framework for representing and decomposing graphs with treewidth at most three, addressing an open problem in graph theory and formal language syntax.

Contribution

It introduces a new syntax generalizing series-parallel expressions and provides a finite set of axioms for graphs of treewidth at most three.

Findings

Finite equational presentation of such graphs

Canonical decomposition into connected components

Finitely many axioms relate all non-deterministic choices

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.