A Finite Presentation of Graphs of Treewidth at Most Three

Authors Amina Doumane, Samuel Humeau, Damien Pous

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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


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

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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)


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.

Subject Classification

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


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  1. Stefan Arnborg, Bruno Courcelle, Andrzej Proskurowski, and Detlef Seese. An algebraic theory of graph reduction. J. ACM, 40(5):1134-1164, 1993. URL: https://doi.org/10.1145/174147.169807.
  2. Stefan Arnborg and Andrzej Proskurowski. Characterization and recognition of partial 3-trees. SIAM Journal on Algebraic Discrete Methods, 7(2):305-314, 1986. URL: https://doi.org/10.1137/0607033.
  3. Umberto Bertelè and Francesco Brioschi. On non-serial dynamic programming. J. Comb. Theory, Ser. A, 14(2):137-148, 1973. URL: https://doi.org/10.1016/0097-3165(73)90016-2.
  4. Hans L. Bodlaender. Treewidth: Algorithmic techniques and results. In Igor Prívara and Peter Ruzicka, editors, Proc. MFCS, volume 1295 of LNCS, pages 19-36. Springer, 1997. URL: https://doi.org/10.1007/BFB0029946.
  5. Enric Cosme-Llópez and Damien Pous. K4-free graphs as a free algebra. In Kim G. Larsen, Hans L. Bodlaender, and Jean-François Raskin, editors, Proc. MFCS, volume 83 of LIPIcs, pages 76:1-76:14. Schloss Dagstuhl, 2017. URL: https://doi.org/10.4230/LIPICS.MFCS.2017.76.
  6. Bruno Courcelle. Graph grammars, monadic second-order logic and the theory of graph minors. In Neil Robertson and Paul D. Seymour, editors, Proc. Graph Structure Theory, volume 147 of Contemporary Mathematics, pages 565-590. American Mathematical Society, 1991. Google Scholar
  7. Bruno Courcelle and Joost Engelfriet. Graph Structure and Monadic Second-Order Logic - A Language-Theoretic Approach, volume 138 of Encyclopedia of mathematics and its applications. Cambridge University Press, 2012. URL: http://www.cambridge.org/fr/knowledge/isbn/item5758776/?site_locale=fr_FR.
  8. Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012. Google Scholar
  9. Christian Doczkal and Damien Pous. Treewidth-two graphs as a free algebra. In Igor Potapov, Paul G. Spirakis, and James Worrell, editors, Proc. MFCS, volume 117 of LIPIcs, pages 60:1-60:15. Schloss Dagstuhl, 2018. URL: https://doi.org/10.4230/LIPICS.MFCS.2018.60.
  10. Amina Doumane, Samuel Humeau, and Damien Pous. A finite presentation of graphs of treewidth at most three, 2024. Version of this paper with the appendix. URL: https://hal.science/hal-04560570.
  11. Rudolf Halin. S-functions for graphs. Journal of geometry, 8:171-186, 1976. URL: https://doi.org/10.1007/BF01917434.
  12. Damien Pous. Hypergraphs. Software, swhId: https://archive.softwareheritage.org/swh:1:dir:028863b2f75dde258591611a6c7c165e289db890;origin=https://github.com/damien-pous/hypergraph;visit=swh:1:snp:76c08bf95d77951c90bdd771a828219ebb4cdcd2;anchor=swh:1:rev:661b5ef93f9d6a2f450f6f0638a1b61780dfe0a4 (visited on 2024-06-17). URL: https://perso.ens-lyon.fr/damien.pous/hypergraph/.
  13. Neil Robertson and Paul D. Seymour. Graph minors. II. algorithmic aspects of tree-width. J. Algorithms, 7(3):309-322, 1986. URL: https://doi.org/10.1016/0196-6774(86)90023-4.
  14. Neil Robertson and Paul D. Seymour. Graph minors. XX. wagner’s conjecture. J. Comb. Theory, Ser. B, 92(2):325-357, 2004. URL: https://doi.org/10.1016/J.JCTB.2004.08.001.
  15. Daniel P. Sanders. Linear algorithms for graphs of tree-width at most four. PhD thesis, Georgia Institute of Technology, 1993. URL: http://hdl.handle.net/1853/30061.
  16. Petra Scheffler. Linear-time algorithms for NP-complete problems restricted to partial k-trees. Technical report, Academy of Sciences of the GDR, 1987. Google Scholar
  17. Thomas V. Wimer. Linear Algorithms on k-Terminal Graphs. PhD thesis, Clemson University, 1993. URL: https://tigerprints.clemson.edu/arv_dissertations/28.