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**Published in:** LIPIcs, Volume 193, 12th International Conference on Interactive Theorem Proving (ITP 2021)

Wagner’s theorem states that a graph is planar (i.e., it can be embedded in the real plane without crossing edges) iff it contains neither 𝖪_5 nor 𝖪_{3,3} as a minor. We provide a combinatorial representation of embeddings in the plane that abstracts from topological properties of plane embeddings (e.g., angles or distances), representing only the combinatorial properties (e.g., arities of faces or the clockwise order of the outgoing edges of a vertex). The representation employs combinatorial hypermaps as used by Gonthier in the proof of the four-color theorem. We then give a formal proof that for every simple graph containing neither 𝖪_5 nor 𝖪_{3,3} as a minor, there exists such a combinatorial plane embedding. Together with the formal proof of the four-color theorem, we obtain a formal proof that all graphs without 𝖪_5 and 𝖪_{3,3} minors are four-colorable. The development is carried out in Coq, building on the mathematical components library, the formal proof of the four-color theorem, and a general-purpose graph library developed previously.

Christian Doczkal. A Variant of Wagner’s Theorem Based on Combinatorial Hypermaps. In 12th International Conference on Interactive Theorem Proving (ITP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 193, pp. 17:1-17:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{doczkal:LIPIcs.ITP.2021.17, author = {Doczkal, Christian}, title = {{A Variant of Wagner’s Theorem Based on Combinatorial Hypermaps}}, booktitle = {12th International Conference on Interactive Theorem Proving (ITP 2021)}, pages = {17:1--17:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-188-7}, ISSN = {1868-8969}, year = {2021}, volume = {193}, editor = {Cohen, Liron and Kaliszyk, Cezary}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2021.17}, URN = {urn:nbn:de:0030-drops-139123}, doi = {10.4230/LIPIcs.ITP.2021.17}, annote = {Keywords: Coq, MathComp, Graph-Theory, Hypermaps, Planarity} }

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**Published in:** LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 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.

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)

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@InProceedings{doczkal_et_al:LIPIcs.MFCS.2018.60, author = {Doczkal, Christian and Pous, Damien}, title = {{Treewidth-Two Graphs as a Free Algebra}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {60:1--60:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.60}, URN = {urn:nbn:de:0030-drops-96429}, doi = {10.4230/LIPIcs.MFCS.2018.60}, annote = {Keywords: Treewidth, Universal Algebra, Rewriting} }

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