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A Finite Presentation of Graphs of Treewidth at Most Three

Authors Amina Doumane, Samuel Humeau, Damien Pous

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Subject Classification

ACM Subject Classification
  • Theory of computation → Equational logic and rewriting
  • Mathematics of computing → Paths and connectivity problems
Keywords
  • Graphs
  • treewidth
  • connectedness
  • axiomatisation
  • series-parallel expressions

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Abstract

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

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Amina Doumane, Samuel Humeau, and Damien Pous. A Finite Presentation of Graphs of Treewidth at Most Three. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 135:1-135:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ICALP.2024.135

Author Details

Amina Doumane
  • Plume, LIP, CNRS, ENS de Lyon, France
Samuel Humeau
  • Plume, LIP, CNRS, ENS de Lyon, France
Damien Pous
  • Plume, LIP, CNRS, ENS de Lyon, France

Funding

This work was supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX- 0007) operated by the French National Research Agency (ANR).

Acknowledgements

The second author would like to thank Ugo Giocanti for several discussions about graph minors and connectivity.

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References

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