My question is: Given a planar graph defined by its adjacency matrix. Can I always find a set of points, so that the dual graph of the voronoi diagram of that set of points is the same as the planar graph?
2026-02-23 03:29:46.1771817386
Is every planar graph a possible dual graph of a voronoi diagram?
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Lemma: given a Voronoi diagram $\mathcal{D}$ and its dual graph $(\mathcal{V}, \mathcal{E})$ consider a "face" of the graph (a bounded connected component of its complement), and the set of all vertices at the boundary of that face. Then all sites of $\mathcal{D}$ corresponding to these vertices lie on the same circle.
Corollary: If the graph below was the dual to a Voronoi diagram, then the three sites corresponding to the vertices in the middle would have to all lie on two different circles, which is impossible. Note that there is not only one way to embed this graph in the plane, but the other ways fail to the same argument.