The polygon diagonal intersection graph 
is the planar graph obtained
from a regular polygon with 
sides by drawing all its diagonals.
Its vertices are the polygon
vertices and the interior intersection points of the
diagonals. Its edges
are the sides of the regular polygon and the segments
into which the intersection points split the diagonals
(Poonen and Rubinstein 1998).
For purposes of extending the vertex- and edge-count sequences below to 
and 2, set 
and 
, where 
is the complete graph.
With these conventions, the vertex counts 
of 
for 
, 2, ... are 1, 2, 3, 5, 10, 19, 42, 57, 135, 171, ... (OEIS
A007569). For 
, these vertex counts
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 
, 4, ..., 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.