Planar and Poly-Arc Lombardi Drawings

TL;DR

This paper introduces k-Lombardi drawings, allowing edges to be composed of multiple arcs, and proves that every planar graph admits a smooth planar 3-Lombardi drawing, expanding the understanding of Lombardi graph representations.

Contribution

It extends Lombardi drawings to k-Lombardi drawings and establishes that all planar graphs can be represented with smooth planar 3-Lombardi drawings.

Findings

Every graph has a smooth 2-Lombardi drawing.

Every planar graph has a smooth planar 3-Lombardi drawing.

Explores the relationship between planarity and Lombardi drawings.

Abstract

In Lombardi drawings of graphs, edges are represented as circular arcs, and the edges incident on vertices have perfect angular resolution. However, not every graph has a Lombardi drawing, and not every planar graph has a planar Lombardi drawing. We introduce k-Lombardi drawings, in which each edge may be drawn with k circular arcs, noting that every graph has a smooth 2-Lombardi drawing. We show that every planar graph has a smooth planar 3-Lombardi drawing and further investigate topics connecting planarity and Lombardi drawings.