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On the Width of Complicated JSJ Decompositions

Authors Kristóf Huszár , Jonathan Spreer

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Author Details

Kristóf Huszár
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Jonathan Spreer
  • School of Mathematics and Statistics F07, The University of Sydney, NSW 2006 Australia

Funding

  • Huszár, Kristóf: Supported by the French National Research Agency through the 3IA Côte d'Azur (ANR-19-P3IA-0002), AlgoKnot (ANR-20-CE48-0007), GrR (ANR-18-CE40-0032), and TWIN-WIDTH (ANR-21-CE48-0014) projects. A previous version of this manuscript was written during the author’s visit at Institut Henri Poincaré (IHP). The author acknowledges the hospitality of IHP (UAR 839 CNRS-Sorbonne Université) and LabEx CARMIN (ANR-10-LABX-59-01).
  • Spreer, Jonathan: Supported by the Australian Research Council’s Discovery funding scheme (project no. DP220102588).

Acknowledgements

We are grateful to Arnaud de Mesmay and to Clément Maria for their interest in our work and for inspiring discussions. We thank the anonymous referees for several suggestions to improve this article.

Cite As Get BibTex

Kristóf Huszár and Jonathan Spreer. On the Width of Complicated JSJ Decompositions. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 42:1-42:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.SoCG.2023.42

Abstract

Motivated by the algorithmic study of 3-dimensional manifolds, we explore the structural relationship between the JSJ decomposition of a given 3-manifold and its triangulations. Building on work of Bachman, Derby-Talbot and Sedgwick, we show that a "sufficiently complicated" JSJ decomposition of a 3-manifold enforces a "complicated structure" for all of its triangulations. More concretely, we show that, under certain conditions, the treewidth (resp. pathwidth) of the graph that captures the incidences between the pieces of the JSJ decomposition of an irreducible, closed, orientable 3-manifold M yields a linear lower bound on its treewidth tw (M) (resp. pathwidth pw(M)), defined as the smallest treewidth (resp. pathwidth) of the dual graph of any triangulation of M.
We present several applications of this result. We give the first example of an infinite family of bounded-treewidth 3-manifolds with unbounded pathwidth. We construct Haken 3-manifolds with arbitrarily large treewidth - previously the existence of such 3-manifolds was only known in the non-Haken case. We also show that the problem of providing a constant-factor approximation for the treewidth (resp. pathwidth) of bounded-degree graphs efficiently reduces to computing a constant-factor approximation for the treewidth (resp. pathwidth) of 3-manifolds.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Geometric topology
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Fixed parameter tractability
Keywords
  • computational 3-manifold topology
  • fixed-parameter tractability
  • generalized Heegaard splittings
  • JSJ decompositions
  • pathwidth
  • treewidth
  • triangulations

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