Drawing Trees with Perfect Angular Resolution and Polynomial Area

TL;DR

This paper investigates methods for drawing trees with perfect angular resolution, demonstrating polynomial area solutions for unordered and Lombardi-style drawings, while highlighting exponential area limitations for ordered straight-line drawings.

Contribution

It introduces new algorithms for crossing-free tree drawings with perfect angular resolution, including polynomial area solutions for unordered and Lombardi-style drawings, and establishes exponential area bounds for ordered straight-line drawings.

Findings

Unordered trees have polynomial area crossing-free drawings with perfect angular resolution.

Ordered trees may require exponential area for straight-line drawings with perfect angular resolution.

Lombardi-style drawings can achieve polynomial area with perfect angular resolution for all trees.

Abstract

We study methods for drawing trees with perfect angular resolution, i.e., with angles at each node v equal to 2{\pi}/d(v). We show: 1. Any unordered tree has a crossing-free straight-line drawing with perfect angular resolution and polynomial area. 2. There are ordered trees that require exponential area for any crossing-free straight-line drawing having perfect angular resolution. 3. Any ordered tree has a crossing-free Lombardi-style drawing (where each edge is represented by a circular arc) with perfect angular resolution and polynomial area. Thus, our results explore what is achievable with straight-line drawings and what more is achievable with Lombardi-style drawings, with respect to drawings of trees with perfect angular resolution.