License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2023.42
URN: urn:nbn:de:0030-drops-178920
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/17892/
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Huszár, Kristóf ; Spreer, Jonathan

On the Width of Complicated JSJ Decompositions

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LIPIcs-SoCG-2023-42.pdf (3 MB)

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.

BibTeX - Entry

@InProceedings{huszar_et_al:LIPIcs.SoCG.2023.42,
  author =	{Husz\'{a}r, Krist\'{o}f and Spreer, Jonathan},
  title =	{{On the Width of Complicated JSJ Decompositions}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{42:1--42:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/17892},
  URN =		{urn:nbn:de:0030-drops-178920},
  doi =		{10.4230/LIPIcs.SoCG.2023.42},
  annote =	{Keywords: computational 3-manifold topology, fixed-parameter tractability, generalized Heegaard splittings, JSJ decompositions, pathwidth, treewidth, triangulations}
}

Keywords: computational 3-manifold topology, fixed-parameter tractability, generalized Heegaard splittings, JSJ decompositions, pathwidth, treewidth, triangulations
Collection: 39th International Symposium on Computational Geometry (SoCG 2023)
Issue Date: 2023
Date of publication: 09.06.2023

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