A note on graph drawings with star-shaped boundaries in the plane

TL;DR

This paper introduces a simple method for embedding planar graphs with a star-shaped boundary using energy minimization, and explores the topological properties of the space of such embeddings.

Contribution

It proposes a new straight-line embedding technique for planar graphs with star-shaped boundaries based on Dirichlet energy minimization and analyzes the contractibility of the embedding space.

Findings

Embedding method based on Dirichlet energy minimization

Contractibility of embedding space for non-convex quadrilaterals

Conjecture for general star-shaped polygons

Abstract

In this note, we propose a straightforward method to produce an straight-line embedding of a planar graph where one face of a graph is fixed in the plane as a star-shaped polygon. It is based on minimizing discrete Dirichlet energies, following the idea of Tutte's embedding theorem. We will call it a geodesic triangulation of the star-shaped polygon. Moreover, we study the homotopy property of spaces of all straight-line embeddings. We give a simple argument to show that this space is contractible if the boundary is a non-convex quadrilateral. We conjecture that the same statement holds for general star-shaped polygons.