CHARACTERIZING DELAUNAY GRAPHS VIA FIXED POINT THEOREM: A SIMPLE PROOF

Abstract

This paper discusses the problem of determining whether a given plane graph is a Delaunay graph, i.e., whether it is topologically equivalent to a Delaunay triangulation. There exist theorems which characterize Delaunay graphs and yield polynomial time algorithms for the problem only by solving some linear inequality systems. A polynomial time algorithm proposed by Hodgson, Rivin and Smith solves a linear inequality system given by Rivin, which is based on sophisticated arguments about hyperbolic geometry. Independently, Hiroshima, Miyamoto and Sugihara gave another linear inequality system and a polynomial time algorithm. Although their discussion is based on primitive arguments on Euclidean geometry, their proofs are long and intricate, unfortunately. In this paper, we give a simple proof of the theorem shown by Hiroshima et al. by employing the fixed point theorem.