On the treewidth of triangulated 3-manifolds

article On the treewidth of triangulated 3-manifolds published yes Kristóf Huszár author 33C26278-F248-11E8-B48F-1D18A9856A870000-0002-5445-5057 Jonathan Spreer author Uli Wagner author 36690CA2-F248-11E8-B48F-1D18A9856A870000-0002-1494-0568 UlWa department In graph theory, as well as in 3-manifold topology, there exist several width-type parameters to describe how "simple" or "thin" a given graph or 3-manifold is. These parameters, such as pathwidth or treewidth for graphs, or the concept of thin position for 3-manifolds, play an important role when studying algorithmic problems; in particular, there is a variety of problems in computational 3-manifold topology - some of them known to be computationally hard in general - that become solvable in polynomial time as soon as the dual graph of the input triangulation has bounded treewidth. In view of these algorithmic results, it is natural to ask whether every 3-manifold admits a triangulation of bounded treewidth. We show that this is not the case, i.e., that there exists an infinite family of closed 3-manifolds not admitting triangulations of bounded pathwidth or treewidth (the latter implies the former, but we present two separate proofs). We derive these results from work of Agol, of Scharlemann and Thompson, and of Scharlemann, Schultens and Saito by exhibiting explicit connections between the topology of a 3-manifold M on the one hand and width-type parameters of the dual graphs of triangulations of M on the other hand, answering a question that had been raised repeatedly by researchers in computational 3-manifold topology. In particular, we show that if a closed, orientable, irreducible, non-Haken 3-manifold M has a triangulation of treewidth (resp. pathwidth) k then the Heegaard genus of M is at most 18(k+1) (resp. 4(3k+1)). https://research-explorer.ista.ac.at/download/7093/7094/479-1917-1-PB.pdf application/pdfno Computational Geometry Laborartoy2019 eng Journal of Computational Geometry 1920-180X 1712.0043410.20382/JOGC.V10I2A5 10270–98 https://research-explorer.ista.ac.at/record/285 https://research-explorer.ista.ac.at/record/8032 Huszár K, Spreer J, Wagner U. On the treewidth of triangulated 3-manifolds. Journal of Computational Geometry. 2019;10(2):70–98. doi:<a href="https://doi.org/10.20382/JOGC.V10I2A5">10.20382/JOGC.V10I2A5</a> Huszár, Kristóf, Jonathan Spreer, and Uli Wagner. “On the Treewidth of Triangulated 3-Manifolds.” Journal of Computational Geometry. Computational Geometry Laborartoy, 2019. <a href="https://doi.org/10.20382/JOGC.V10I2A5">https://doi.org/10.20382/JOGC.V10I2A5</a>. Huszár, Kristóf, et al. “On the Treewidth of Triangulated 3-Manifolds.” Journal of Computational Geometry, vol. 10, no. 2, Computational Geometry Laborartoy, 2019, pp. 70–98, doi:<a href="https://doi.org/10.20382/JOGC.V10I2A5">10.20382/JOGC.V10I2A5</a>. Huszár, K., Spreer, J., & Wagner, U. (2019). On the treewidth of triangulated 3-manifolds. Journal of Computational Geometry. Computational Geometry Laborartoy. <a href="https://doi.org/10.20382/JOGC.V10I2A5">https://doi.org/10.20382/JOGC.V10I2A5</a> Huszár K, Spreer J, Wagner U. 2019. On the treewidth of triangulated 3-manifolds. Journal of Computational Geometry. 10(2), 70–98. K. Huszár, J. Spreer, U. Wagner, Journal of Computational Geometry 10 (2019) 70–98. K. Huszár, J. Spreer, and U. Wagner, “On the treewidth of triangulated 3-manifolds,” Journal of Computational Geometry, vol. 10, no. 2. Computational Geometry Laborartoy, pp. 70–98, 2019. 70932019-11-23T12:14:09Z2026-04-08T07:21:27Z