Treewidth-Two Graphs as a Free Algebra

Authors Christian Doczkal, Damien Pous

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Author Details

Christian Doczkal
  • Univ Lyon, CNRS, ENS de Lyon, UCB Lyon 1, LIP, France
Damien Pous
  • Univ Lyon, CNRS, ENS de Lyon, UCB Lyon 1, LIP, France

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Christian Doczkal and Damien Pous. Treewidth-Two Graphs as a Free Algebra. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 60:1-60:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


We give a new and elementary proof that the graphs of treewidth at most two can be seen as a free algebra. This result was originally established through an elaborate analysis of the structure of K_4-free graphs, ultimately reproving the well-known fact that the graphs of treewidth at most two are precisely those excluding K_4 as a minor. Our new proof is based on a confluent and terminating rewriting system for term-labeled graphs and does not involve graph minors anymore. The new strategy is simpler and robust in the sense that it can be adapted to subclasses of treewidth-two graphs, e.g., graphs without self-loops.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Theory of computation → Rewrite systems
  • Computing methodologies → Symbolic and algebraic manipulation
  • Treewidth
  • Universal Algebra
  • Rewriting


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