License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2019.44
URN: urn:nbn:de:0030-drops-104487
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10448/
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Huszár, Kristóf ; Spreer, Jonathan

3-Manifold Triangulations with Small Treewidth

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LIPIcs-SoCG-2019-44.pdf (0.9 MB)

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.

BibTeX - Entry

@InProceedings{huszr_et_al:LIPIcs:2019:10448,
  author =	{Krist{\'o}f Husz{\'a}r and Jonathan Spreer},
  title =	{{3-Manifold Triangulations with Small Treewidth}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{44:1--44:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Gill Barequet and Yusu Wang},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10448},
  URN =		{urn:nbn:de:0030-drops-104487},
  doi =		{10.4230/LIPIcs.SoCG.2019.44},
  annote =	{Keywords: computational 3-manifold topology, fixed-parameter tractability, layered triangulations, structural graph theory, treewidth, cutwidth, Heegaard genus}
}

Keywords: computational 3-manifold topology, fixed-parameter tractability, layered triangulations, structural graph theory, treewidth, cutwidth, Heegaard genus
Collection: 35th International Symposium on Computational Geometry (SoCG 2019)
Issue Date: 2019
Date of publication: 11.06.2019

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