Degree sequences of graphs and dominance order

Suppose that the graphical partition H(A) = (a: 2 . . . 2 a:) arises from A = (al 2 . . . 2 a,) by deleting the largest summand a1 from A and reducing the a1 largest of the remaining summands by one. Let (a;+l 2 . . 2 ah) = H(A) denote the partition obtained by applying the operator H i times. We prove that the dominance order of partitions is preserved when we switch from A to (a1 2 a: 2 . 2 2 ' .) =: €(A). This generalizes a recent result by Favaron, Maheo, and Sacle on the residue of a graph.