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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 $T$?

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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.

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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