Consider the plane figure obtained by drawing each diagonal in a regular polygon with 
vertices. If each point of intersection is associated with
a node and diagonals are split ar each intersection to form segments associated with
edges, the resulting figure is a planar graph here
termed the polygon diagonal intersection graph and denoted 
.
For 
,
2, ..., the vertex counts 
of 
are 1, 2, 3, 5, 10, 19, 42, 57, 135, 171, ... (OEIS A007569),
which are given by a finite sum of
 |
(1)
|
times polynomials in 
with 
, 4, 6, 12, 18, 24, 30, 42, 60, 84, 90, 120, and 210 (Poonen
and Rubinstein 1998).
For 
,
2, ..., the edge counts 
of 
are 0, 1, 3, 8, 20, 42, 91, 136, 288, ... (OEIS A135565),
which are again given by a finite sum of polynomials times 
.
Similarly, for 
, 2, ..., the numbers of regions 
into which the polygon is divided are given by 1, 4, 11,
24, 50, 80, 154, 220, 375, ... (OEIS A007678),
where the 
th term is given in closed form by
 |
(2)
|
For 
odd, all terms but the first drop out, so the numbers of regions are given by
 |
(3)
|
Precomputed properties of polygonal diagonal intersection graphs are implemented in the Wolfram Language as GraphData[
"DiagonalIntersection",
n
].
-Gon With Diagonals: Bounds on Area of Largest Cell?"
Dec. 17, 2022. 
." J. Int. Seq. 13, No. 10.8.5,
2010.