Wikipedia on Delaunay Triangulation says:
"In the plane[...], if there are b vertices on the convex hull, then any
triangulation of the points has at most 2n − 2 − b triangles, plus one
exterior face."
Here n is the total number of vertices in the plane.
I can't see how they obtained this result when I use Euler's formula 1=V-E+F. So V=n then F=1-n+E. what is face exterior?
2 Answers
Euler's formular for planar graphs with $n$ vertices, $e$ edges and $f$ faces says that $2 = n - e + f$. Note that $f$ includes the outer, unbounded face to infinity, too. So we have $f-1$ triangles and a $b$-gon.
We can assign to each edge two tokens; one for each incident face. The number of tokens can be counted per face, too: $3(f-1)$ for $f-1$ triangles and $b$ for the outer face, so $3 (f-1) + b = 2e$.
Then we have $4 = 2n - 2e + 2f = 2n -f -k + 3$, and hence $f-1 = 2n -2 -b$, which is the number of triangles.
The exterior face is the green area.
