A Finite Algorithm for the Realizabilty of a Delaunay Triangulation

The \emph{Delaunay graph} of a point set $P\subseteq {\mathbb{R}}^2$ is the plane graph with the vertex-set $P$ and the edge-set that contains $\{p,p^{\mathrm{\prime }}\}$ if there exists a disc whose intersection with $P$ is exactly $\{p,p^{\mathrm{\prime }}\}$. Accordingly, a triangulated graph $G$ is \emph{Delaunay realizable} if there exists a triangulation of the Delaunay graph of some $P\subseteq {\mathbb{R}}^2$, called a \emph{Delaunay triangulation} of $P$, that is isomorphic to $G$. The objective of $\text{textsc}DelaunayRealization$ is to compute a point set $P\subseteq {\mathbb{R}}^2$ that realizes a given graph $G$ (if such a $P$ exists). Known algorithms do not solve $\text{textsc}DelaunayRealization$ as they are non-constructive. Obtaining a constructive algorithm for $\text{textsc}DelaunayRealization$ was mentioned as an open problem by Hiroshima et al.~\cite{hiroshima2000}. We design an $n^{\mathcal{O}(n)}$-time constructive algorithm for $\text{textsc}DelaunayRealization$. In fact, our algorithm outputs sets of points with {\em integer} coordinates.