Efficient algorithms for domination and Hamilton circuit problems on permutation graphs
 P. Shanthi Sastry,
 N. Jayakumar,
 C. E. Veni Madhavan
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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 NPComplete. 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.
 A. Brandstadt and D. Kratsch, "On the restriction of some NPcomplete Graph Problems to Permutation Graphs", Proc. of Foundations of Computation Theory '85, Lecture Notes in Computer Science 199, Springer, Berlin 1985.
 E.J. Cockayne and S.T. Hedetniemi, "Towards a Theory of Domination in Graphs", Networks, 7(1977) pp.247–261.
 M. Farber and J.M. Keil, "Domination in Permutation Graphs", J. Algorithms, 6(1985) pp.309–321.
 M.R. Garey and D.S. Johnson, "Computers and Intractability: A Guide to the Theory of NPcompleteness," W.H. Freeman and Co., New York 1977.
 M.C. Golumbic, “Algorithmic Graph Theory and Perfect Graphs”, Academic Press, New York, 1980.
 O.Kariv and S.L. Hakimi, "An Algorithmic approach to Network Location Problems 1. The pcenters", SIAM J. Appl. Math., 37(1979)
 D.S. Johnson, "NPcompleteness Column: An Ongoing guide", J. Algorithms, 6(1985) pp.434–451. CrossRef
 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. CrossRef
 Title
 Efficient algorithms for domination and Hamilton circuit problems on permutation graphs
 Book Title
 Foundations of Software Technology and Theoretical Computer Science
 Book Subtitle
 Seventh Conference, Pune, India December 17–19, 1987 Proceedings
 Pages
 pp 6578
 Copyright
 1987
 DOI
 10.1007/3540186255_43
 Print ISBN
 9783540186250
 Online ISBN
 9783540480334
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 287
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag
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 Editors
 Authors

 P. Shanthi Sastry ^{(1)}
 N. Jayakumar ^{(1)}
 C. E. Veni Madhavan ^{(1)}
 Author Affiliations

 1. Dept. of Computer Science and Automation, Indian Institute of Science, 560 012, Bangalore, India
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