While research on the geometry of planar graphs has been active in the past decades, many properties of planar metrics remain mysterious. This paper studies a fundamental aspect of the planar graph geometry: covering planar metrics by a small collection of simpler metrics. Specifically, a \emph{tree cover} of a metric space $(X, \delta)$ is a collection of trees, so that every pair of points $u$ and $v$ in $X$ has a low-distortion path in at least one of the trees. The celebrated ``Dumbbell Theorem'' [ADMSS95] states that any low-dimensional Euclidean space admits a tree cover with $O(1)$ trees and distortion $1+\varepsilon$, for any fixed $\varepsilon \in (0,1)$. This result has found numerous algorithmic applications, and has been generalized to the wider family of doubling metrics [BFN19]. Does the same result hold for planar metrics? A positive answer would add another evidence to the well-observed connection between Euclidean/doubling metrics and planar metrics. In this work, we answer this fundamental question affirmatively. Specifically, we show that for any given fixed $\varepsilon \in (0,1)$, any planar metric can be covered by $O(1)$ trees with distortion $1+\varepsilon$. Our result for planar metrics follows from a rather general framework: First we reduce the problem to constructing tree covers with \emph{additive distortion}. Then we introduce the notion of \emph{shortcut partition}, and draw connection between shortcut partition and additive tree cover. Finally we prove the existence of shortcut partition for any planar metric, using new insights regarding the grid-like structure of planar graphs. [...]

翻译：尽管过去几十年对平面图几何性质的研究十分活跃，但平面度量的许多性质仍不为人知。本文研究平面图几何的一个基本方面：用少量更简单的度量覆盖平面度量。具体而言，度量空间 $(X, \delta)$ 的**树覆盖**是一组树的集合，使得 $X$ 中任意一对点 $u$ 和 $v$ 在至少一棵树中存在一条低失真路径。著名的“哑铃定理”[ADMSS95]指出，对任意固定的 $\varepsilon \in (0,1)$，任何低维欧几里得空间均存在由 $O(1)$ 棵树组成的树覆盖，且失真度为 $1+\varepsilon$。该结果已有大量算法应用，并被推广至更广泛的加倍度量族[BFN19]。同样的结果是否适用于平面度量？肯定的回答将为欧几里得/加倍度量与平面度量之间已被广泛观察到的联系增添又一证据。在本文中，我们肯定地回答了这一基本问题。具体而言，我们证明：对任意给定的固定 $\varepsilon \in (0,1)$，任何平面度量均可被 $O(1)$ 棵树以失真度 $1+\varepsilon$ 覆盖。我们关于平面度量的结果源于一个相当通用的框架：首先将问题简化为构造具有**加性失真**的树覆盖。接着引入**捷径划分**概念，并建立捷径划分与加性树覆盖之间的联系。最后，利用关于平面图网格状结构的新见解，证明任何平面度量均存在捷径划分。[…]