Graphic and Protographic Lists of Integers - Fon-Der-Flaass, Dmitry; West, Douglas B. (The Electronic Journal of Combinatorics, 2004)
A positive list (list of positive integers) is protographic if its merger with all but finitely many positive graphic lists is graphic. Define the family P_s of s-protogaphic lists by letting P_0 be the family of positive graphic lists and letting P_s for s>0 be the family of positive lists whose merger with all but finitely many lists in P_s-1 is in P_s-1. The main result is that XP_s if and only if t(X)P_s-1, where t(X) is the list obtained from X by subtracting one from each term of X (deleting those that become 0) and appending a 1 for each term of X. A corollary is that the maximum number of iterations to reach a graphic list from an n-term even list with sum 2k is k-n+1 (when kn), achieved by the unique such list having one term larger than 1.