On an edge ranking problem of trees and graphs

A k-edge ranking of an undirected graph is a labeling of the edges of the graph with integers 1, 2, ..., k, with the property that all paths between two edges with the same label i contain an edge with label j[rang]i. The edge ranking problem is that of finding the smallest k for which a graph has a k-edge ranking. This problem is useful in the optimization of the number of parallel stages required to assemble a product from its components. The problem is also related to that of finding minimum height edge partition trees of graphs. The main result in the paper is an O(n log n) time approximation algorithm for edge ranking of trees, which has a worst case performance ratio of 2. © 1991.