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Discussiones Mathematicae Graph Theory 26(1) (2006) 149-159
DOI: https://doi.org/10.7151/dmgt.1309

OPTIMAL EDGE RANKING OF COMPLETE BIPARTITE GRAPHS IN POLYNOMIAL TIME

Ruo-Wei Hung

Department of Information Management
Nan-Kai Institute of Technology
Tsao-Tun, Nantou 542, Taiwan
e-mail: rwhung@cs.ccu.edu.tw

Abstract

An edge ranking of a graph is a labeling of edges using positive integers such that all paths connecting two edges with the same label visit an intermediate edge with a higher label. An edge ranking of a graph is optimal if the number of labels used is minimum among all edge rankings. As the problem of finding optimal edge rankings for general graphs is NP-hard [12], it is interesting to concentrate on special classes of graphs and find optimal edge rankings for them efficiently. Apart from trees and complete graphs, little has been known about special classes of graphs for which the problem can be solved in polynomial time. In this paper, we present a polynomial-time algorithm to find an optimal edge ranking for a complete bipartite graph by using the dynamic programming strategy.

Keywords: graph algorithms, edge ranking, vertex ranking, edge-separator tree, complete bipartite graphs.

2000 Mathematics Subject Classification: 05C78, 05C85, 68Q25.

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Received 17 June 2005
Revised 20 October 2005