Discrete Math

There is an algorithm to treat this problem
See, Buckley F., and Harary F. (1990), Distance in graphs p.167
Form the sequence in descending order
(4,4,4,2,2,2)
Step 1 : Delete the first term 4
Step2 : Substract 1 from each of the next 4 terms, gives (3,3,1,1,2)
Step3 : Rearrange the sequence to (3,3,2,1,1)
Step4 : Delete the first term 3
Step5 : Substract 1 from each of the next 3 terms, gives (2,1,0,1)
Step6 : Rearrange the sequence to (2,1,1,0)
Step 7 : Delete the first term 2
Step 8 : Substract 1 from each of the next 2 terms, gives (0,0,0)
Since all terms are 0, we conclude that there exist a simple graph G=(V,E) with degree sequences as follows:
(2,2,2,4,4,4)