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COMMENT
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For n=9 we have the following valid graphic partitions; 9,82,73,64,55,443,533,542,632,722,3333,4332,4422,5322,43222. The basic pattern is partitions of n+k into k+1 parts, minimum part 2. After checking a graph can be produced (e.g. 6222 cannot), adding the number of distinct elements in each pattern gives the sequence, except for (n-1)2, which is always 1 and only counting elements which are greater than or equal to the number of elements in a pattern (e.g. 722 only yields 1 possibility). So the patterns above yield 1,1,2,2,1,2,2,3,3,1,1,2,1,2,1, adding gives a(9)=25
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