Suppose I have a collection of $n$ non-collinear points on a sphere, $\left\lbrace P_i\right\rbrace_{i=1}^n $. And I construct a mapping from this collection of points to the Delaunay Triangulation on the sphere, $\left\lbrace P_i\right\rbrace_{i=1}^n \overset{DT}{\mapsto} \left( \left\lbrace P_i\right\rbrace_{i=1}^n, \mathcal{F}_{\left\lbrace P_i\right\rbrace_{i=1}^n} , \mathcal{E}_{\left\lbrace P_i\right\rbrace_{i=1}^n}\right) $, where
- $\left\lbrace P_i\right\rbrace_{i=1}^n$ is the set of vertices in the delaunay triangulation.
- $\mathcal{F}_{\left\lbrace P_i\right\rbrace_{i=1}^n}$ is the set of faces (lets just say the faces are the interior regions only) on the sphere.
- $\mathcal{E}_{\left\lbrace P_i\right\rbrace_{i=1}^n} $ is the set of edges. Edges are really arc segments on the sphere.
To illustrate the point, here is an example of such a Delaunay triangulation:
Kulikowski's Theorem states, "For every positive integer, $n$, there exists a sphere which has exactly $n$ lattice points on its surface."
So lets call these $n$ lattice points, $\left\lbrace L_i\right\rbrace_{i=1}^n$ and corresponding Delaunay Triangulation mapping as such $\left\lbrace L_i\right\rbrace_{i=1}^n \overset{DT}{\mapsto} \left( \left\lbrace L_i\right\rbrace_{i=1}^n, \mathcal{F}_{\left\lbrace L_i\right\rbrace_{i=1}^n} , \mathcal{E}_{\left\lbrace L_i\right\rbrace_{i=1}^n}\right) $
So, I have some questions, and they are in descending ordered in which I desire an answer for:
- Is there a graph isomorphism from $\left( \left\lbrace P_i\right\rbrace_{i=1}^n, \mathcal{F}_{\left\lbrace P_i\right\rbrace_{i=1}^n} , \mathcal{E}_{\left\lbrace P_i\right\rbrace_{i=1}^n}\right) $ to $\left( \left\lbrace L_i\right\rbrace_{i=1}^n, \mathcal{F}_{\left\lbrace L_i\right\rbrace_{i=1}^n} , \mathcal{E}_{\left\lbrace L_i\right\rbrace_{i=1}^n}\right) $? If so, does it preserve the (Face-Edge) relationships between the graphs?
- If a graph isomorphism exists, is there a way to explicitly define the graph isomorphic mappings?
- What is the basis for this lattice on the sphere for Kulikowski's Theorem? Is it any arbitrary basis?
I'm not really sure how to cite the image. I got it from this paper: http://hal.inria.fr/docs/00/44/19/38/PDF/RR-7004.pdf