ON-LINE VERTEX RANKING OF TREES

A k-ranking of a graph G is a labeling of its vertices from {1, . . ., k} such that any nontrivial path whose endpoints have the same label contains a larger label. The least k for which G has a k-ranking is the ranking number of G, also known as tree-depth. Applications of rankings include VLSI design, parallel computing, and factory scheduling. The on-line ranking problem asks for an algorithm to rank the vertices of G as they are revealed one at a time in the subgraph of G induced by the vertices revealed so far (each previously revealed vertex appears with its label, but the final placement of the induced subgraph in G is not specified). The on-line ranking number of G is the minimum over all such algorithms of the largest label that algorithm can be forced to use. We give algorithmic bounds on the on-line ranking number of trees in terms of maximum degree, diameter, and number of internal vertices.