When does a graph admit a tree-decomposition in which every bag has small diameter? For finite graphs, this is a property of interest in algorithmic graph theory, where it is called having bounded “tree-length”. We will show that this is equivalent to being “boundedly quasi-isometric to a tree”, which for infinite graphs is a much-studied property from metric geometry. One object of this paper is to tie these two areas together. We will prove that there is a tree-decomposition in which each bag has small diameter, if and only if there is a map \(\phi \) from V(G) into the vertex set of a tree T, such that for all \(u,v\in V(G)\), the distances \(d_G(u,v), d_T(\phi (u),\phi (v))\) differ by at most a constant. A necessary condition for admitting such a tree-decomposition is that there is no long geodesic cycle, and for graphs of bounded tree-width, Diestel and Müller showed that this is also sufficient. But it is not sufficient in general, even qualitatively, because there are graphs in which every geodesic cycle has length at most three, and yet every tree-decomposition has a bag with large diameter. There is a more general necessary condition, however. A “geodesic loaded cycle” in G is a pair (C, F), where C is a cycle of G and \(F\subseteq E(C)\), such that for every pair u, v of vertices of C, one of the paths of C between u, v contains at most \(d_G(u,v)\) F-edges, where \(d_G(u,v)\) is the distance between u, v in G. We will show that a (possibly infinite) graph G admits a tree-decomposition in which every bag has small diameter, if and only if |F| is small for every geodesic loaded cycle (C, F). Our proof is an extension of an algorithm to approximate tree-length in finite graphs by Dourisboure and Gavoille. In metric geometry, there is a similar theorem that characterizes when a graph is quasi-isometric to a tree, “Manning’s bottleneck criterion”. The goal of this paper is to tie all these concepts together, and add a few more related ideas. For instance, we prove a conjecture of Rose McCarty, that G admits a tree-decomposition in which every bag has small diameter, if and only if for all vertices u, v, w of G, some ball of small radius meets every path joining two of u, v, w.

图表什么时候允许树分解，其中每个袋子的直径都很小？对于有限图，这是算法图论中一个有趣的属性，它被称为有界“树长度”。我们将证明这相当于“有界准等距到一棵树”，对于无限图来说，这是度量几何中一个被广泛研究的属性。本文的一个目标是将这两个领域联系在一起。我们将证明存在树分解，其中每个袋具有较小的直径，当且仅当存在从V ( G ) 到树T的顶点集的映射\(\phi \)，使得对于所有\(u,v\in V(G)\)，距离\(d_G(u,v), d_T(\phi (u),\phi (v))\)最多相差一个常数。承认这种树分解的必要条件是不存在长测地线循环，并且对于有界树宽度的图，Diestel 和 Müller 表明这也是足够的。但一般来说，甚至在定性上也是不够的，因为在有些图中，每个测地线循环的长度最多为三，但每个树分解都有一个大直径的袋子。然而，还有一个更普遍的必要条件。G中的“测地线负载循环”是一对 ( C ,  F )，其中C是G和\(F\subseteq E(C)\)的循环，这样对于C的每对顶点u、  v，u ,  v之间的C路径之一最多包含\(d_G(u,v)\) F边，其中\(d_G(u,v)\)是G中u ,  v之间的距离。我们将证明一个（可能是无限的）图G允许树分解，其中每个袋子都有小直径，当且仅当 | F |对于每个测地加载循环 ( C ,  F ) 来说都很小。我们的证明是 Dourisboure 和 Gavoille 提出的一种算法的扩展，用于近似有限图中的树长度。在度量几何中，有一个类似的定理来表征图与树的拟等距，即“曼宁瓶颈准则”。本文的目标是将所有这些概念联系在一起，并添加一些更多相关的想法。例如，我们证明 Rose McCarty 的猜想，即G承认树分解，其中每个袋子都有小直径，当且仅当对于G的所有顶点u、  v、  w ，某个小半径的球会遇到连接u、  v、  w中的两个的每条路径。