On the L(h,k)-Labeling of Co-comparability Graphs

Abstract

Given two non negative integers h and k, an L(h,k)-labeling of a graph G = (V,E) is a map from V to a set of labels such that adjacent vertices receive labels at least h apart, while vertices at distance at most 2 receive labels at least k apart. The goal of the L(h,k)-labeling problem is to produce a legal labeling that minimizes the largest label used. Since the decision version of the L(h,k)-labeling problem is NP-complete, it is important to investigate classes of graphs for which the problem can be solved efficiently.

Along this line of though, in this paper we deal with co-comparability graphs and two of its subclasses: interval graphs and unit-interval graphs. Specifically, we provide, in a constructive way, the first upper bounds on the L(h,k)-number of co-comparability graphs and interval graphs. To the best of our knowledge, ours is the first reported result concerning the L(h,k)-labeling of co-comparability graphs.

In the special case where k = 1, our result improves on the best previously-known approximation ratio for interval graphs.

This research was supported, in part, by the European Research Project Algorithmic Principles for Building Efficient Overlay Computers (AEOLUS). Most of the work reported here was performed while Professor Olariu visited with the Department of Computer Science, University of Rome “La Sapienza”. Support through a Visiting Fellowship from the University of Rome “La Sapienza” is gratefully acknowledged.