Tutte's celebrated barycentric embedding theorem describes a natural way to build straight-line embeddings (crossing-free drawings) of a (3-connected) planar graph: map the vertices of the outer face to the vertices of a convex polygon, and ensure that each remaining vertex is in convex position, namely, a barycenter with positive coefficients of its neighbors. Actually computing an embedding then boils down to solving a system of linear equations. A particularly appealing feature of this method is the flexibility given by the choice of the barycentric weights. Generalizations of Tutte's theorem to surfaces of nonpositive curvature are known, but due to their inherently continuous nature, they do not lead to an algorithm. In this paper, we propose a purely discrete analog of Tutte's theorem for surfaces (with or without boundary) of nonpositive curvature, based on the recently introduced notion of reducing triangulations. We prove a Tutte theorem in this setting: every drawing homotopic to an embedding such that each vertex is harmonious (a discrete analog of being in convex position) is a weak embedding (arbitrarily close to an embedding). We also provide a polynomial-time algorithm to make an input drawing harmonious without increasing the length of any edge, in a similar way as a drawing can be put in convex position without increasing the edge lengths.

翻译：Tutte著名的重心嵌入定理描述了一种构建（3-连通）平面图直线嵌入（无交叉绘制）的自然方法：将外表面的顶点映射到凸多边形的顶点，并确保每个剩余顶点处于凸位置，即其邻接点的具有正系数的重心。实际计算嵌入可归结为求解线性方程组。该方法特别吸引人的特点在于重心权重选择提供的灵活性。Tutte定理已推广至非正曲率曲面，但由于其固有的连续性本质，这些推广并未形成算法。本文基于最近提出的约化三角剖分概念，针对非正曲率曲面（带边界或无边界）提出了纯离散的Tutte定理模拟。我们在此框架下证明了Tutte定理：每个与嵌入同伦且满足各顶点调和（凸位置的离散模拟）的绘制都是弱嵌入（无限逼近于嵌入）。我们还提供了一种多项式时间算法，可在不增加任何边长度的前提下使输入绘制达到调和状态，其原理类似于将绘制调整为凸位置而不增加边长度。