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arXiv · 2508.18535

Tangling and Untangling Trees on Point-sets

Giuseppe Di Battista·Giuseppe Liotta·Maurizio Patrignani·Antonios Symvonis·Ioannis G. Tollis
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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 $\fracπ{2}$.

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Giuseppe Di Battista, Giuseppe Liotta, Maurizio Patrignani, Antonios Symvonis, Ioannis G. Tollis. 2025-08-25. Tangling and Untangling Trees on Point-sets. https://arxiv.org/abs/2508.18535

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