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Graph Theory in Coq: Minors, Treewidth, and Isomorphisms

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Abstract

We present a library for graph theory in Coq/Ssreflect. This library covers various notions on simple graphs, directed graphs, and multigraphs. We use it to formalize several results from the literature: Menger’s theorem, the excluded-minor characterization of treewidth-two graphs, and a correspondence between multigraphs of treewidth at most two and terms of certain algebras.

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Notes

  1. To be fully precise, we use the type \(\varSigma x:G.\,x \in U = \textsf {true}\), exploiting that set membership is decidable. Since equality between booleans is proof irrelevant (i.e., there is at most one proof of \(x \in U = \textsf {true}\)), this ensures that \(\left| (G|_U)\right| = |\varSigma x:G.\,x \in U| = |U|\). This is a standard technique used pervasively in the mathematical components library.

  2. Note that \(\overline{V_2} = V_1 \setminus V_2\) if \( V_1 \cup V_2 = V\).

  3. To be precise, the functional may call its argument on anything. However, the result may only depend on calls to smaller arguments in the domain.

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Acknowledgements

We would like to thank Guillaume Combette, with whom we developed the first version of the library. We are also grateful to Nicolas Trotignon for his wonderful insights on graph theory.

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Authors and Affiliations

  1. Inria Sophia Antipolis Méditerranée, Université Côte d’Azur, Nice, France

    Christian Doczkal

  2. CNRS, ENS de Lyon, UCB Lyon 1, LIP, Univ Lyon, Lyon, France

    Damien Pous

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Correspondence to Christian Doczkal.

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Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This paper extends and revises the results presented in [10]; the underlying Coq library is available from https://perso.ens-lyon.fr/damien.pous/covece/graphs/.

This work has been funded by the European Research Council (ERC) under the European Union’s Horizon 2020 programme (CoVeCe, Grant Agreement No. 678157), and was supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon and \(\hbox {UCA}^{\text {JEDI}}\), within the programs “Investissements d’Avenir” ANR-11-IDEX-0007 and ANR-15-IDEX-01, respectively.

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Doczkal, C., Pous, D. Graph Theory in Coq: Minors, Treewidth, and Isomorphisms. J Autom Reasoning 64, 795–825 (2020). https://doi.org/10.1007/s10817-020-09543-2

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