Planar Lombardi Drawings for Subcubic Graphs

David Eppstein

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arXiv · 1206.6142

Planar Lombardi Drawings for Subcubic Graphs

David Eppstein

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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.

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David Eppstein. 2012-06-26. Planar Lombardi Drawings for Subcubic Graphs. https://arxiv.org/abs/1206.6142

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