Planar Lombardi Drawings for Subcubic Graphs

TL;DR

This paper proves that all planar graphs with maximum degree three can be drawn with circular edges meeting at equal angles, using circle packing and hyperbolic geometry, and provides an algorithm for such drawings.

Contribution

It introduces a novel circle packing-based method for Lombardi drawings of subcubic planar graphs, extending the class of graphs known to admit such drawings.

Findings

Every subcubic planar graph has a Lombardi drawing.

The method is based on circle packings and hyperbolic geometry.

Not all 4-regular planar graphs admit Lombardi drawings.

Abstract

We prove that every planar graph with maximum degree three has a planar drawing in which the edges are drawn as circular arcs that meet at equal angles around every vertex. Our construction is based on the Koebe-Thurston-Andreev circle packing theorem, and uses a novel type of Voronoi diagram for circle packings that is invariant under Moebius transformations, defined using three-dimensional hyperbolic geometry. We also use circle packing to construct planar Lombardi drawings of a special class of 4-regular planar graphs, the medial graphs of polyhedral graphs, and we show that not every 4-regular planar graph has a Lombardi drawing. We have implemented our algorithm for 3-connected planar cubic graphs.