On Domination Problems for Permutation and Other Graphs

Abstract There is an increasing interest in results on the influence of restricting NP-complete graph problems to special classes of perfect graphs as, e.g., permutation graphs. It was shown that several problems restricted to permutation graphs are solvable in polynomial time [2, 3, 4, 6, 7, 14, 16]. In this paper we give 1. (i) an algorithm with time bound O ( n 2 ) for the weighted independent domination problem on permutation graphs (which is an improvement of the O ( n 3 ) solution given in [7]); 2. (ii) a polynomial time solution for the weighted feedback vertex set problem on permutation graphs; 3. (iii) an investigation of (weighted) dominating clique problems for several graph classes including an NP-completeness result for weakly triangulated graphs as well as polynomial time bounds.

[1]  Jeremy P. Spinrad.  Transitive orientation in 0(n2) time , 1983, STOC '83.

[2]  David S. Johnson.  The NP-Completeness Column: An Ongoing Guide , 1986, J. Algorithms.

[3]  Alfred B. Manaster,et al.  Two-Dimensional Partial Orderings: Recursive Model Theory , 1980, J. Symb. Log..

[4]  Yehoshua Perl,et al.  Clustering and domination in perfect graphs , 1984, Discret. Appl. Math..

[5]  Dieter Kratsch,et al.  On the restriction of some NP-complete graph problems to permutation graphs , 1985, FCT.

[6]  David S. Johnson,et al.  The NP-Completeness Column: An Ongoing Guide , 1982, J. Algorithms.

[7]  Lorna Stewart.  Permutation graph structure and algorithms , 1985 .

[8]  Jeremy P. Spinrad,et al.  Bipartite permutation graphs , 1987, Discret. Appl. Math..

[9]  Claude Berge,et al.  Graphs and Hypergraphs , 2021, Clustering.

[10]  A. Lempel,et al.  Transitive Orientation of Graphs and Identification of Permutation Graphs , 1971, Canadian Journal of Mathematics.

[11]  Martin Farber,et al.  Domination in Permutation Graphs , 1985, J. Algorithms.

[12]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[13]  J. Keil,et al.  Finding minimum dominating cycles in permutation graphs , 1985 .

[14]  Ryan B. Hayward,et al.  Weakly triangulated graphs , 1985, J. Comb. Theory B.

[15]  Amir Pnueli,et al.  Permutation Graphs and Transitive Graphs , 1972, JACM.

[16]  M. Golumbic.  Algorithmic graph theory and perfect graphs , 1980 .