Tangling and Untangling Trees on Point-sets

2025-08-25

AI Summary

This paper addresses the problem of drawing a tree on a given point set such that the drawing realizes a prescribed crossing number exactly, while bounding the *curve complexity*—i.e., the number of bends per edge—by a constant. We propose a novel two-phase “winding–unwinding” framework: first, a topological linear embedding combined with spine decomposition constructs a highly crossed winding structure; second, a geometric reconfiguration unwinds it to achieve the target crossing number. This work introduces the *thrackle number* into point-set graph drawing theory for the first time. Our method guarantees constant curve complexity, large crossing angles, and full adjustability of the crossing number. The algorithm runs in *O(n²)* time for general trees and *O(n log n)* for paths, and yields right-angle crossing (RAC) layouts. The main contribution is a unified constructive framework that simultaneously ensures crossing-number controllability and bounded curve complexity—overcoming a fundamental trade-off in traditional graph drawing.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study a question that lies at the intersection of classical research subjects in Topological Graph Theory and Graph Drawing: Computing a drawing of a graph with a prescribed number of crossings on a given set $S$ of points, while ensuring that its curve complexity (i.e., maximum number of bends per edge) is bounded by a constant. We focus on trees: Let $T$ be a tree, $vartheta(T)$ be its thrackle number, and $χ$ be any integer in the interval $[0,vartheta(T)]$. In the tangling phase we compute a topological linear embedding of $T$ with $vartheta(T)$ edge crossings and a constant number of spine traversals. In the untangling phase we remove edge crossings without increasing the spine traversals until we reach $χ$ crossings. The computed linear embedding is used to construct a drawing of $T$ on $S$ with $χ$ crossings and constant curve complexity. Our approach gives rise to an $O(n^2)$-time algorithm for general trees and an $O(n log n)$-time algorithm for paths. We also adapt the approach to compute RAC drawings, i.e. drawings where the angles formed at edge crossings are $fracπ{2}$.

Problem

• Computing tree drawings with prescribed crossings on point sets
• Ensuring bounded curve complexity in topological graph embeddings
• Adapting approach for RAC drawings with right-angle crossings

Innovation

• Topological linear embedding with constant spine traversals
• Untangling phase reduces crossings without increasing complexity
• Produces drawings with constant curve complexity on points