# Efficient algorithms for domination and Hamilton circuit problems on permutation graphs

## Abstract

The domination and Hamilton circuit problems are of interest both in algorithm design and complexity theory. The domination problem has applications in facility location and the Hamilton circuit problem has applications in routing problems in communications and operations research.

The problem of deciding if G has a dominating set of cardinality at most k, and the problem of determining if G has a Hamilton circuit are NP-Complete. Polynomial time algorithms are, however, available for a large number of restricted classes. A motivation for the study of these algorithms is that they not only give insight into the characterization of these classes but also require a variety of algorithmic techniques and data structures. So the search for efficient algorithms, for these problems in many classes still continues.

A class of perfect graphs which is practically important and mathematically interesting is the class of permutation graphs. The domination problem is polynomial time solvable on permutation graphs. Algorithms that are already available are of time complexity O(n^{2}) or more, and space complexity O(n^{2}) on these graphs. The Hamilton circuit problem is open for this class.

We present a simple O(n) time and O(n) space algorithm for the domination problem on permutation graphs. Unlike the existing algorithms, we use the concept of geometric representation of permutation graphs. Further, exploiting this geometric notion, we develop an O(n^{2}) time and O(n) space algorithm for the Hamilton circuit problem.

## Preview

Unable to display preview. Download preview PDF.

### References

- [1]A. Brandstadt and D. Kratsch, "On the restriction of some NP-complete Graph Problems to Permutation Graphs", Proc. of Foundations of Computation Theory '85, Lecture Notes in Computer Science 199, Springer, Berlin 1985.Google Scholar
- [2]E.J. Cockayne and S.T. Hedetniemi, "Towards a Theory of Domination in Graphs", Networks, 7(1977) pp.247–261.Google Scholar
- [3]M. Farber and J.M. Keil, "Domination in Permutation Graphs", J. Algorithms, 6(1985) pp.309–321.Google Scholar
- [4]M.R. Garey and D.S. Johnson, "Computers and Intractability: A Guide to the Theory of NP-completeness," W.H. Freeman and Co., New York 1977.Google Scholar
- [5]M.C. Golumbic, “Algorithmic Graph Theory and Perfect Graphs”, Academic Press, New York, 1980.Google Scholar
- [6]O.Kariv and S.L. Hakimi, "An Algorithmic approach to Network Location Problems 1. The p-centers", SIAM J. Appl. Math., 37(1979)Google Scholar
- [7]D.S. Johnson, "NP-completeness Column: An Ongoing guide", J. Algorithms, 6(1985) pp.434–451.CrossRefGoogle Scholar
- [8]K.J. Supowit, "Decomposing A set of Points into Chains with an Application to Permutation and Circle Graphs", Info. Proc.Letters, 21(1985) pp. 249–252.CrossRefGoogle Scholar