Characterizing Delaunay Graphs via Fixed Point Theorem

This paper discusses a problem for determining whether a given plane graph is a Delaunay graph, i.e., whether it is topologically equivalent to a Delaunay triangulation. There exists a theorem which characterizes Delaunay graphs and yields a polynomial time algorithm for the problem only by solving a certain linear inequality system. The theorem was proved by Rivin based on arguments of hyperbolic geometry. Independently, Hiroshima, Miyamoto and Sugihara gave another proof of the theorem based on primitive arguments on Euclidean geometry. Unfortunately, the existing proofs of the theorem are rather difficult or long. In this paper, we give a simple proof of the theorem characterizing Delaunay