3-Manifold Triangulations with Small Treewidth

Huszár, Kristóf; Spreer, Jonathan

doi:10.4230/lipics.socg.2019.44

Cited by 1 publication

(2 citation statements)

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“…The theoretical efficiency of the aforementioned FPT algorithms crucially depends on the assumption that the dual graph of the input triangulation has small treewidth. To understand their scope, it is thus instructive to consider the treewidth tw(M) of a compact 3-manifold M, defined as the smallest treewidth of the dual graph of any triangulation of M. Indeed, the relation between the treewidth and other quantities associated with 3-manifolds has recently been investigated in various contexts [25,26,27,28,41]. For instance, in [28] together with Wagner we have shown that the treewidth of a non-Haken 3-manifold is always bounded below in terms of its Heegaard genus.…”

Section: Introductionmentioning

confidence: 99%

“…Assume that we glue the manifolds M v along the arcs of G to obtain a closed 3-manifold M. Without restrictions on how these pieces are glued together, this cannot result in a lower bound tw(M) in terms of tw(G): we can construct Seifert fibered spaces M in this way, even if G = (V, E) is the complete graph with |V | arbitrarily large. At the same time, Seifert fibered spaces have constant treewidth, see [27].…”

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On the width of complicated JSJ decompositions

Huszár¹,

Spreer²

2023

Preprint

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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.

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