Minimal Edge Covers in the n-Sun Graph
======================================

The n-sun graph has n outside vertices and n inside vertices.

Illustration of sun graph of order 4 and a possible minimal edge cover:

           o                        o
         /   \                    /   \
        o-----o                  o     o
      / |\   /| \
     o  |  X  |  o             o         o
      \ |/   \| /               \       /
        o-----o                  o     o
         \   /                        /
           o                        o

The inside n vertices of the sun graph are connected internally as a complete graph.
The number of minimal edge covers of a complete graph is enumerated by A053530(n).

An edge covering is a set of edges with each graph vertex being incident with at least one edge of the covering.
In a minimal edge covering the edges may only be stars. (The subgraph o--o--o--o is forbidden)

To enumerate the number of minimal edge covers we first consider the possible coverings of the outside vertices
and then consider the number of ways to complete the inside vertices for each outside configuration.

A covering of the outside edges can be represented by the following symbolisms.
    /\    - outside vertex connected to both neighboring vertices
    \/    - two neighboring outside vertices connected to the same inside vertex
     o    - disconnected inside vertex
     /    - outside vertex connected to inside vertex to left
     \    - outside vertex connected to inside vertex to right

All outside vertices must be connected so no symbolism is required to represent a disconnected outside vertex.

Reading clockwise from the top vertex, the covering of the 4-sun graph in the illustration above can be represented linearly:
  /\ \/ / 

Let k be the number of /\ elements in the configuration.
Let j be the number of o elements in the configuration.

For every /\ element there must be a \/ element so 3*k <= n.
For every o element there must be a \/ element so 3*k + 2*j <= n.

The case of k=0 and k>0 will be considered separately
Case k=0:
This is the case that each outside vertex is connected to exactly one inside vertex.
Each can be connected to the left or right without restriction so in total there are 2^n possible configurations.
For n=3 the 8 possibilties are:
     / / /    \ \ \
     / o \/  \/ o \
     o \/ /  \ \/ o
     o \ \/  \/ / o
The generaral counting rule in terms of j is to start with all outside vertices connected either to the left or the right and
then change diretion 2j times in binomial(n, 2*j) ways.

For example for n=3, j=0 => 2*binomial(n,2*j) = 2; 
                     j=1 => 2*binomial(n,2*j) = 6.

Case k>0:
First start the configuration with k pairs of /\ \/ elements.
For example for k=2 the starting point is /\ \/ /\ \/
Then insert into the configuration j pairs of o \/ following any of the k \/ elements.
This can be done in binomial(j+k-1, j) ways.
Then insert into the configuration n-3*k-2*j / or \ elements following any of the 2*k+2*j elements
to bring the configuration to the correct length. The choice of whether to insert a / or \ is determined
by the preceeding element. (a \ follows /\ or \ and / follows o)
This can be done in binomial(n-k-1,2*k+2*j-1) ways.
The above only counts configurations starting with /\. To get the total number it is necessary to multiply by n/k.

Example for n=6. There are a total of 9 configurations starting with /\ as follows:
    k=2, j=0:     /\ \/ /\ \/
    k=1, j=1:     /\ \/ o \/ /,  /\ \/ o \ \/,  /\ \/ / o \/,  /\ \ \/ o \/
    k=1, j=0:     /\ \/ / / /,   /\ \ \/ / /,   /\ \ \ \/ /,   /\ \ \ \ \/

We now need to examine the possibilities for adding the internal edges of the inner n-complete graph to the edge cover.
Firstly inside edges may only be used to connect a o to a o, \/, / or \.
All other connection possibilities break the minimal edge cover. (a non star will be formed).
Also every o element must be connected to create an edge cover.
The number of 'o' elments is j and the number of \/, / or \ is n-2*k-j.
Some of the j 'o' elements may be connected to only other 'o' elements. Let i be this number. (0 <= i <= j).
These i elements form a complete subgraph of order i which has A053530(i) minimal edge covers.
The remaining j-i 'o' elements are then assigned arbitrarily to the n-2*k-j other elements.
This can be done in (n-2*k-j)^(j-i) ways.

Let b(i) = A053530(i)               
                ___
               \        
Let c(p,j) =    >       binomial(j,i) * b(i) * p^(j-i)
               /___
               i=0..j

Then c(n-2*k-j,j) counts the number of possibilities for the internal edges of the inner complete graph.

               ___
              \
Let a(n)  =    >    2*binomial(n,2*j) * c(n-j, j) 
              /___ 
               j           
                 ___    ___
                \      \      n 
             +   >      >     - * binomial(j+k-1, j) * binomial(n-k-1, 2*k+2*j-1) * c(n-2*k-j, j)
                /___   /___   k
                 k>0    j

Then a(n) is the number of minimal edge covers in the n-sun graph.