A (multi)set of segments in the plane may form a TSP tour, a matching, a tree, or any multigraph. If two segments cross, then we can reduce the total length with the following flip operation. We remove a pair of crossing segments, and insert a pair of non-crossing segments, while keeping the same vertex degrees. The goal of this paper is to devise efficient strategies to flip the segments in order to obtain crossing-free segments after a small number of flips. Linear and near-linear bounds on the number of flips were only known for segments with endpoints in convex position. We generalize these results, proving linear and near-linear bounds for cases with endpoints that are not in convex position. Our results are proved in a general setting that applies to multiple problems, using multigraphs and the distinction between removal and insertion choices when performing a flip.

翻译：平面中的（多重）线段集合可构成旅行商问题路径、匹配、树或任意多重图。若两条线段交叉，可通过以下翻转操作减少总长度：移除一对交叉线段，插入一对非交叉线段，同时保持顶点度数不变。本文旨在设计高效策略，通过少量翻转操作实现线段无交叉。现有结果仅针对端点处于凸位置的线段，证明了翻转次数呈线性或近线性界。我们将这些结果推广至端点非凸位置的情形，证明了线性与近线性界。本文在适用于多类问题的通用框架下完成证明，该框架利用多重图，并区分执行翻转时移除与插入选择的不同。