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Treewidth-Two Graphs as a Free Algebra

Abstract

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 presented last year at MFCS and was proved through an elaborate analysis of the structure of K4-free graphs, ultimately reproving the well-known fact that the graphs of treewidth at most two are precisely those excluding K4 as a minor. Our new proof is based on a confluent and terminating rewriting system for treewidth-two 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. It could also be amenable to handle graphs of larger treewidth, where the excluded minors are typically unknown.
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Dates and versions

hal-01780844 , version 1 (28-04-2018)
hal-01780844 , version 2 (12-09-2018)
hal-01780844 , version 3 (12-09-2018)

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Christian Doczkal, Damien Pous. Treewidth-Two Graphs as a Free Algebra. Mathematical Foundations of Computer Science, Aug 2018, Liverpool, United Kingdom. ⟨10.4230/LIPIcs.MFCS.2018.60⟩. ⟨hal-01780844v3⟩
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