A Delaunay triangulation is simply the dual graph
for a Voronoi tessellation [8,9].
Furthermore, the dual graph
of [SIGMA] [F.sub.i] must be a tree.
We improve the dual graph
technique introduced by Anez et al.
The experiments using different gene expression data sets for testing have made comparatively ideal experimental results, which proves the validity of the dual graph
For instance, to study curves on a general K3 surface, we can let it degenerate to a union of projective planes, the dual graph
of which is a triangulation of the real 2-sphere.We shall consider the following kind of families of subvarieties: families of curves with prescribed invariants and singularities in surfaces (with special attention to the two cases of the projective plane, and of K3 surfaces), families of hyperplane sections with prescribed singularities of hypersurfaces in projective spaces, families of curves with a given genus in Calabi-Yau threefolds, and families of surfaces in the projective 3-space containing curves with unexpected singularities.
Every plane graph has a dual graph
, formed by assigning a vertex of , to each face of and joining two vertices of by edges if and only if the corresponding faces of share edges in their boundaries.
Let G(F, E, W) be the dual graph
of M(V, F) where F is the nodes of the dual graph
, E is the edge set of the dual graph
, each edge connects two neighboring faces, and W is the weights defined on edges.
To construct the posets X (k, d), we adopt a technique that is related to the theory of manifold crystallizations, which will allow us to present a simplicial poset in terms of an edge-labeled dual graph
. Ferri et al.
 introduced the concept of a fuzzy dual graph
and discussed some of its interesting properties.
Consider the dual graph
D of T, which contains one vertex for each region that is bounded by edges of T and the boundary of the convex hull (excluding the region outside the convex hull of G).
Let [G.sup.*.sub.x,y] be the geometric dual graph
of the plane graph [G.sub.0] - x - y.
The four regions 00, 01, 10 and 11 in that figure are the nodes of the dual graph
[ILLUSTRATION FOR FIGURE 1 OMITTED] [C.sub.4], the 4-cycle, also known as the 2-dimensional hypercube [Q.sub.2].