On-Line Vertex Ranking of Trees

Abstract

A $k$-ranking of a graph $G$ is a labeling of its vertices from $\{1,\ldots,k\}$ 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.