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.

翻译：受三维流形算法研究的启发，我们探讨了给定三维流形的JSJ分解与其三角剖分之间的结构关系。基于Bachman、Derby-Talbot和Sedgwick的工作，我们证明了一个“足够复杂”的三维流形JSJ分解会强制其所有三角剖分具有“复杂结构”。更具体地，我们证明在某些条件下，捕获不可约闭可定向三维流形M的JSJ分解中各片之间关联的图的树宽（或路径宽），可给出其树宽tw(M)（或路径宽pw(M)）的线性下界，其中tw(M)（或pw(M)）定义为M的任意三角剖分的对偶图的最小树宽（或路径宽）。我们给出了该结果的若干应用。我们首次给出了具有有界树宽但无界路径宽的三维流形无穷族。我们构造了具有任意大树宽的Haken三维流形；此前仅知道非Haken情形下存在此类流形。我们还证明，将有界度图的树宽（或路径宽）的常数因子逼近问题高效地归约为计算三维流形的树宽（或路径宽）的常数因子逼近问题。