Maximal directed paths in the join digraph
for independent sets of the order 4 graphs.

emptyset -> 1 
            2 
            3 
            4 
            5 
     3,4 -> 6 
            7 
            8 
     4,5 -> 9 
   4,5,7 -> 9 
     4,7 -> 9 
     5,6 -> 9 
   5,6,7 -> 9 
     5,7 -> 9 
     6,7 -> 9 
  5,7,8 -> 10 
    5,8 -> 10 
    7,8 -> 10 
    8,9 -> 10 
           11