On-line vertex ranking of trees
Abstract
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 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.