Voronoi diagram with given number of vertices and sites

Question

I want to draw a Voronoi diagram with 9 sites and with

1. no vertex,
2. 1 vertex,
3. 4 vertices, and
4. 7 vertices.

How do I approach this question. The one with no vertex is easy, it can be done by collinear points. What about the others.

A figure for each would be appreciated.

Answer 1

Conceptually, it is maybe easier to construct a Delaunay triangulation tessalation. The DT is the dual to the Voronoi diagramm, so you want to limit the number of faces to $0,1,4,7$. This can be achieved by placing some of the vertices of the DT on a circle, or by picking the right number of vertices on the convex hull.

Here is a picture for the 4 vertex case (black DT, red Voronoi diagramm). Play around to get the solutions of 2. and 4. (You already solved 1.)