Untangling Segments in the Plane

TL;DR

This work introduces a unifying flip-based framework for untangling crossing planar segments, parameterized by a point-property $\Pi$ and a graph-property $\Gamma$, and quantified by $D_{\Pi,\Gamma}(n)$ and related variants that capture worst-case flip counts under different choices. It systematically analyzes four regimes—no choice, removal choice, insertion choice, and both choices—across several geometric graph problems (TSP, Matching, RedBlue Matching, Tree, Multigraph) and configurations ranging from convex to near-convex, establishing bounds that interpolate between linear, near-linear, and quadratic behavior. Central contributions include reductions tying different graph-properties, near-convex bounds $D(n,t)=O(tn^2)$ for the Matching version, and a suite of phase-driven strategies yielding $O(n \log n)$ and $O(tn \log n)$ results in convex and separated settings; a subcubic bound on distinct flips $O(n^{8/3})$ is also established for the no-choice setting. These results unify disparate prior bounds, show how removal and insertion choices propagate across problems, and illuminate how convexity and separations influence untangling efficiency, with practical implications for local-search heuristics in Euclidean TSP and related reconfiguration problems.

Abstract

A set of n segments in the plane may form a Euclidean TSP tour, a tree, or a matching, among others. Optimal TSP tours as well as minimum spanning trees and perfect matchings have no crossing segments, but several heuristics and approximation algorithms may produce solutions with crossings. 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. In this paper, we consider the number of flips performed under different assumptions, using a new unifying framework that applies to tours, trees, matchings, and other types of (multi)graphs. Within this framework, we prove several new bounds that are sensitive to whether some endpoints are in convex position or not.

Figures (22)

• Figure 1: Examples of flips in a (a) TSP tour, (b) tree, (c) monochromatic matching, (d) multigraph, and (e) red-blue matching.
• Figure 2: The (a) flip graph and a (b) removal strategy of a five-point set in the TSP version.
• Figure 3: (a) The flip graph $F$ of a six-point set in the matching version. (b) Removal, (c) insertion, and (d) both removal and insertion strategies for the flip graph $F$. Pairs of edges corresponding to the same crossing are grouped.
• Figure 4: (a) A crossing-free multigraph transformed into a matching by replacing multi-degree points by clusters of points. (b) Two flips in the $\mathtt{RedBlue}$ version simulating one flip in the $\mathtt{Matching}$ version. (c) Two flips in the $\mathtt{TSP}$ version simulating one flip in the $\mathtt{RedBlue}$ version. (d) Two flips in the $\mathtt{Tree}$ version simulating one flip in the $\mathtt{RedBlue}$ version.
• Figure 5: (a) An $f$-critical line $\ell$ for a flip $f$ removing $p_1p_3,p_2p_4$ and inserting $p_1p_4,p_2p_3$. This situation corresponds to case (2a) with $\ell \in L_1$. (b) An $f$-dropping line $\ell$. This situation corresponds to case (2b) with $\ell \in L_2$.

Theorems & Definitions (44)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4: BoM16FGR24
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• Lemma 3.4