Consider a graph drawn on a surface (for example, the plane minus a finite set of obstacle points), possibly with crossings. We provide an algorithm to decide whether such a drawing can be untangled, namely, if one can slide the vertices and edges of the graph on the surface (avoiding the obstacles) to remove all crossings; in other words, whether the drawing is homotopic to an embedding. While the problem boils down to planarity testing when the surface is the sphere or the disk (or equivalently the plane without any obstacle), the other cases have never been studied before, except when the input graph is a cycle, in an abundant literature in topology and more recently by Despr\'e and Lazarus [SoCG 2017, J. ACM 2019]. Our algorithm runs in O(m + poly(g+b) n log n) time, where g >= 0 and b >= 0 are the genus and the number of boundary components of the input orientable surface S, and n is the size of the input graph drawing, lying on some fixed graph of size m cellularly embedded on S. We use various techniques from two-dimensional computational topology and from the theory of hyperbolic surfaces. Most notably, we introduce reducing triangulations, a novel discrete analog of hyperbolic surfaces in the spirit of systems of quads by Lazarus and Rivaud [FOCS 2012] and Erickson and Whittlesey [SODA 2013], which have the additional benefit that reduced paths are unique and stable upon reversal; they are likely of independent interest. Tailored data structures are needed to achieve certain homotopy tests efficiently on these triangulations. As a key subroutine, we rely on an algorithm to test the weak simplicity of a graph drawn on a surface by Akitaya, Fulek, and T\'oth [SODA 2018, TALG 2019].

翻译：考虑绘制在曲面（例如，去掉有限个障碍点的平面）上的图，可能带有交叉。我们提出一种算法，用于判断此类图形是否能被解开，即能否在曲面上滑动图的顶点和边（避开障碍物）以移除所有交叉；换言之，该图形是否同伦于一个嵌入。当曲面为球面或圆盘（等价于无障碍的平面）时，问题简化为平面性检验，但其他情况此前从未被研究过，唯一的例外是输入图为环的情形——该情形在拓扑学文献中已有大量研究，并近期由Despré和Lazarus [SoCG 2017, J. ACM 2019] 进一步探讨。我们的算法运行时间为 O(m + poly(g+b) n log n)，其中 g≥0 和 b≥0 分别为输入可定向曲面 S 的亏格和边界分支数，n为输入图的图形大小，该图形位于某个大小为 m 且细胞嵌入于 S 的固定图上。我们运用了二维计算拓扑学及双曲曲面理论中的多种技术。尤为重要的是，我们引入了“约化三角剖分”——这是一种双曲曲面的新型离散模拟，灵感来源于Lazarus和Rivaud [FOCS 2012] 以及Erickson和Whittlesey [SODA 2013] 的“四边系统”思想，其额外优势在于约化路径在反转时具有唯一性和稳定性；这一概念可能具有独立的研究价值。为实现对这些三角剖分的高效同伦测试，需要定制化的数据结构。作为关键子程序，我们依赖于Akitaya、Fulek和Tóth [SODA 2018, TALG 2019] 提出的用于测试曲面上图形弱简单性的算法。