Graphic and Protographic Lists of Integers

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 Ps of s-protogaphic lists by letting P0 be the family of positive graphic lists and letting Ps for s > 0 be the family of positive lists whose merger with all but finitely many lists in Ps−1 is in Ps−1. The main result is that X ∈ Ps if and only if t(X) ∈ Ps−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 k ≥ n), achieved by the unique such list having one term larger than 1.