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(n²) or more, and space complexity O(n²) 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(n²) time and O(n) space algorithm for the Hamilton circuit problem.