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

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.

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