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3-Manifold Triangulations with Small Treewidth

Authors Kristóf Huszár , Jonathan Spreer

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Author Details

Kristóf Huszár
  • Institute of Science and Technology Austria (IST Austria), Am Campus 1, 3400 Klosterneuburg, Austria
Jonathan Spreer
  • Institut für Mathematik, Freie Universität Berlin, Arnimallee 2, 14195 Berlin, Germany

Funding

  • Huszár, Kristóf: Partially supported by the Einstein Foundation Berlin (grant EVF-2015-230) for his visits at Freie Universität Berlin.
  • Spreer, Jonathan: Supported by grant EVF-2015-230 of the Einstein Foundation Berlin as well as by the DFG Collaborative Research Center SFB/TRR 109 "Discretization in Geometry and Dynamics".

Acknowledgements

We thank the developers of the free software Regina [Burton, 2013; Burton et al., 1999] for creating a fantastic tool, and the anonymous reviewers for useful comments and suggestions regarding the exposition. KH thanks the people at the Discrete Geometry Group, Freie Universität Berlin, for their hospitality.

Cite AsGet BibTex

Kristóf Huszár and Jonathan Spreer. 3-Manifold Triangulations with Small Treewidth. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 44:1-44:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.SoCG.2019.44

Abstract

Motivated by fixed-parameter tractable (FPT) problems in computational topology, we consider the treewidth tw(M) of a compact, connected 3-manifold M, defined to be the minimum treewidth of the face pairing graph of any triangulation T of M. In this setting the relationship between the topology of a 3-manifold and its treewidth is of particular interest. First, as a corollary of work of Jaco and Rubinstein, we prove that for any closed, orientable 3-manifold M the treewidth tw(M) is at most 4g(M)-2, where g(M) denotes Heegaard genus of M. In combination with our earlier work with Wagner, this yields that for non-Haken manifolds the Heegaard genus and the treewidth are within a constant factor. Second, we characterize all 3-manifolds of treewidth one: These are precisely the lens spaces and a single other Seifert fibered space. Furthermore, we show that all remaining orientable Seifert fibered spaces over the 2-sphere or a non-orientable surface have treewidth two. In particular, for every spherical 3-manifold we exhibit a triangulation of treewidth at most two. Our results further validate the parameter of treewidth (and other related parameters such as cutwidth or congestion) to be useful for topological computing, and also shed more light on the scope of existing FPT-algorithms in the field.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Geometric topology
  • Theory of computation → Fixed parameter tractability
Keywords
  • computational 3-manifold topology
  • fixed-parameter tractability
  • layered triangulations
  • structural graph theory
  • treewidth
  • cutwidth
  • Heegaard genus
  • lens spaces
  • Seifert fibered spaces

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