  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 $\{p,p'\}$
if there exists a disc whose intersection with $P$ is exactly $\{p,p'\}$.
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 \textsc{Delaunay Realization} 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 \textsc{Delaunay Realization} as
they are non-constructive. Obtaining a constructive algorithm for
\textsc{Delaunay Realization} was mentioned as an open problem by Hiroshima et
al.~\cite{hiroshima2000}. We design an $n^{\mathcal{O}(n)}$-time constructive
algorithm for \textsc{Delaunay Realization}. In fact, our algorithm outputs
sets of points with {\em integer} coordinates.
