List rankings and on-line list rankings of graphs

A kk-ranking   of a graph GG is a labeling of its vertices from {1,…,k}{1,…,k} such that any path on at least two vertices whose endpoints have the same label contains a larger label. The least kk for which GG has a kk-ranking is the ranking number   of GG, also known as tree-depth. The list ranking number   of GG is the least kk such that if each vertex of GG is assigned a set of kk potential labels, then GG can be ranked by labeling each vertex with a label from its assigned list. Rankings model a certain parallel processing problem in manufacturing, while the list ranking version adds scheduling constraints. We compute the list ranking number of paths, cycles, and trees with many more leaves than internal vertices. Some of these results follow from stronger theorems we prove about on-line versions of list ranking, where each vertex starts with an empty list having some fixed capacity, and potential labels are presented one by one, at which time they are added to the lists of certain vertices; the decision of which of these vertices are actually to be ranked with that label must be made immediately.