Tangling and Untangling Trees on Point-sets
Giuseppe Di Battista, Giuseppe Liotta, Maurizio Patrignani, Antonios Symvonis, Ioannis G. Tollis

TL;DR
This paper presents algorithms for drawing trees on point sets with a prescribed number of crossings and bounded curve complexity, bridging topological graph theory and graph drawing.
Contribution
It introduces an efficient method to compute tree drawings with a specified number of crossings and constant curve complexity, including RAC drawings.
Findings
O(n^2) algorithm for general trees
O(n log n) algorithm for paths
Extension to RAC drawings with right-angle crossings
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 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 be a tree, be its thrackle number, and be any integer in the interval . In the tangling phase we compute a topological linear embedding of with 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 crossings. The computed linear embedding is used to construct a drawing of on with crossings and constant curve complexity. Our approach gives rise to…
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