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推动，我们探究给定三维流形的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，我们证明三维流形的“足够复杂”的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，刻画不可约、闭、可定向三维流形M的JSJ分解各块之间关联关系的图的树宽（resp. 路径宽）给出了其树宽tw(M)（resp. 路径宽pw(M)）的一个线性下界，其中tw(M)（resp. pw(M)）定义为M的任意三角剖分的对偶图的最小树宽（resp. 路径宽）。我们给出了该结果的若干应用。我们首次构造了一个无穷有界树宽三维流形族，其路径宽无界。我们构造了具有任意大树宽的Haken三维流形；此前这类三维流形的存在性仅在非Haken情形下已知。我们还证明，将有界度图的树宽（resp. 路径宽）的常数因子近似问题可高效约简为计算三维流形的树宽（resp. 路径宽）的常数因子近似问题。