A partition
is called graphical if there exists a graph having degree sequence . The number of graphical
partitions of length is equal to the number of -node graphs that have no isolated
points.

The numbers of distinct graphical partitions corresponding to graphs on , 2, ... nodes are 0, 1, 2, 7, 20, 71, 240, 871, 3148, ...
(OEIS A095268).

A graphical partition of order is one for which the sum of degrees is . A -graphical partition only exists for even .

It is possible for two topologically distinct graphs to have the same degree
sequence, an example of which is illustrated above.

The numbers of graphical partitions on , 4, 6, ... edges are 1, 2, 5, 9, 17, 31, 54, 90, 151, 244,
... (OEIS A000569).