An O(N+M)-Time Algorithm for Finding a Minimum-Weight Dominating Set in a Permutation Graph

Farber and Keil [ Algorithmica, 4 (1989), pp. 221--236] presented an $O(n^3)$-time algorithm for finding a minimum-weight dominating set in permutation graphs. This result was improved to $O(n^2 \log^2n)$ by Tsai and Hsu [SIGAL '90 Algorithms, Lecture Notes in Computer Science, Springer-Verlag, New York, 1990, pp. 109--117] and to $O(n(n + m))$ by the authors of this paper [ Inform. Process. Lett., 37 (1991), pp. 219--224], respectively. In this paper, we introduce a new faster algorithm that takes only $O(n + m)$ time to solve the same problem, where $m$ is the number of edges in a graph of $n$ vertices.