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:在对三维元件的算法研究的推动下,我们探索了JSJ将给定的三维元件分解成像及其三角体之间的结构关系。在Bachman、Derby-Talbot和Sedgwick的工作基础上,我们展示了JSJ将三维元件分解成“足够复杂”的“JSJ”将它的所有三角体的“复杂结构”强化为“最小的树维度 ” 。更具体地说,我们展示了在某些条件下, JSJ的树维度(resp. roadwith) 及其三角体的分解成像。我们给出了JSJJ在不可降、封闭、可调整的三维元件之间分解的成像事件之间事件之间的结构结构关系。我们给出了第一个例子,在Orent-We-wiwi 中提供了一条无限的、不固定的三维系的路径,在先前的轨道上展示了一条不固定的路径。</s>