On-line vertex ranking of trees

A $k$-ranking of a graph $G$ is a labeling of its vertices from $\{1,\dots ,k\}$ 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 presented one at a time along with all previously ranked vertices and the edges between them (so each vertex is presented as the lone unranked vertex in a partially labeled induced subgraph of $G$ whose final placement 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 bounds on the on-line ranking number of trees in terms of maximum degree, diameter, and number of interior vertices.