<p>Yes. See</p>

<p>Dillencourt, Michael B.; Smith, Warren D. Graph-theoretical conditions for inscribability and Delaunay realizability. <a href="http://dx.doi.org/10.1016/0012-365X%2895%2900276-3">Discrete Math. 161 (1996), no. 1-3, 63–77</a>. </p>

<p>However, if a topological triangulation has no separating triangles, and no chords connecting pairs of vertices on its outer face, then it is always realizable as a Delaunay triangulation.</p>

CC BY-SA 3.0
2014-05-12T18:06:00.307Z
2014-05-12T18:06:00.307Z
David Eppstein
<p>Do Delaunay triangulations satisfy any extra topological conditions over normal planar triangulations?</p>

<p>In other words, if $T$ is a topological triangulation of the plane, when does there exist points $x_i \in \mathbb{R}^2$ s.t. their Delaunay triangulation has topology $T$?</p>

<cg.comp-geom><delaunay-triangulation>
Topological properties of Delaunay triangulations
Geoffrey Irving
# Topological properties of Delaunay triangulations

(Asked by: username_0 on 2014-05-12)

Do Delaunay triangulations satisfy any extra topological conditions over normal planar triangulations?

In other words, if $T$ is a topological triangulation of the plane, when does there exist points $x\_i \in \mathbb{R}^2$ s.t. their Delaunay triangulation has topology $T$?

**Tags:** cg.comp-geom, delaunay-triangulation

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## 1 answers

### Answer by username_1 on 2014-05-12. Score: 7 | Accepted answer

Yes. See

Dillencourt, Michael B.; Smith, Warren D. Graph-theoretical conditions for inscribability and Delaunay realizability. [Discrete Math. 161 (1996), no. 1-3, 63–77](http://dx.doi.org/10.1016/0012-365X%2895%2900276-3).

However, if a topological triangulation has no separating triangles, and no chords connecting pairs of vertices on its outer face, then it is always realizable as a Delaunay triangulation.


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