The triangulations generators


A triangulation of the plane  or equivalently of the sphere 
is a plane graph with all faces triangles. Because these triangulations are the
most common ones, they are simetimes just called "triangulation" without explicitly
mentioning the sphere or plane.
There is also one window to generate triangulations of the disk. In this window
there is one distinguished outer face  the outside of the disk  that need not be a triangle.
For a lot of these structures the current  chemically oriented  3Dembedders do not produce nice
pictures. Writing specialized embedders for these structures will be future work.

General Triangulations
In this window you can
generate triangulations without too many restrictions very efficiently.
The number of vertices, the
minimum degree of the vertices, and a lower bound for the connectivity number must be given. Due to the Euler formula and the fact that the graphs are simple the only
possible values for the connectivity number of a triangulation with at least 4 vertices are 3,4, and 5.
As an option one can choose to only generate graphs with connectivity number exactly the lower bound.


Eulerian Triangulations
A graph is called Eulerian if it admits an Eulertour  that is a closed walk using
each edge exactly once. Such a closed walk exists if and only if the degree of every vertex is
even.
The number of vertices must be chosen, but due to the Euler formula the minimum degree of a simple triangulation with at least 4 vertices is always 4  so no choice is possible. The connectivity number can be 3 or 4
and it is possible to choose a lower bound or the exact value of the connectivity number of the graphs
generated.


Triangulations with given vertex degrees
In this window you can generate triangulations in which the vertices have degrees out of
some given list.
The required parameters are the number of vertices and a list of allowed degrees.
Optionally lower and upper bounds for the number of vertices for each degree can
be specified.


Triangulations of the Disk
Triangulations of the disk have a distinguished outer face  the outside of the disk  which
is not required to be a triangle. Isomorphisms must map these distinguished outer faces onto
each other.
In cases where the distinguished face is larger than a triangle, two
triangulations of the disk are isomorphic as plane graphs if and only if they
are isomorphic as triangulations of the disk. In cases where the outer face is a triangle, it is possible
that several isomorphic plane graphs are generated  but there will be no isomorphisms mapping
the distinguished faces of different graphs onto each other. In order to emphasize this important role of the
distinguished face, it is always displayed as the outer face in the 2Ddrawings.
The number of vertices of the triangulation of the disk must be given. Furthermore
it is possible to
 fix the size of the distinguished face
 allow, forbid, or even require chords (that is: edges not in the boundary of the distinguished face but connecting vertices in the distinguised face)
 allow and forbid 2valent vertices in the boundary. Note that 2valent vertices not in the
boundary are not possible for simple triangulations of the disk.


Further Links
 Stuart Anderson has used plantri and later CaGe to generate graphs
for his collection of Squared
Rectangles.
The mathematics
and implementation of plantri are a collaboration between Gunnar Brinkmann
and Brendan D. McKay, who distributes the "naked" generator
on his website.




