10.4230/LIPIcs.SoCG.2019.44
Huszár, Kristóf
Kristóf
Huszár0000-0002-5445-5057
Spreer, Jonathan
Jonathan
Spreer
3-manifold triangulations with small treewidth
Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik
2019
2019-06-11T20:09:57Z
2019-08-02T12:39:22Z
conference
https://research-explorer.app.ist.ac.at/record/6556
https://research-explorer.app.ist.ac.at/record/6556.json
978-3-95977-104-7
1868-8969
1812.05528
905885 bytes
application/pdf
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.