Labeling algorithms for domination problems in sun-free chordal graphs

Abstract A k-dominating set of a graph G=(V,E) is a set of vertices D such that for every vertex x in V there exists some vertex y in D satisfying d(x,y)≤k. A k-dominating set D of G is connected if the subgraph G[D] induced by D is connected and total if G[D] has no isolated vertex. This paper presents efficient algorithms for finding a minimum cardinality k-dominating set without taking power, connected k-dominating set and total 1-dominating set of a sun-free chordal graph. NP-complete results for these problems are also discussed.

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

[2]  S. M. Hedetniemi,et al.  On the Algorithmic Complexity of Total Domination , 1984 .

[3]  Kellogg S. Booth,et al.  Dominating Sets in Chordal Graphs , 1982, SIAM J. Comput..

[4]  D. Corneil,et al.  A dynamic programming approach to the dominating set problem on k trees , 1987 .

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

[6]  George L. Nemhauser,et al.  R-domination on block graphs , 1982, Oper. Res. Lett..

[7]  Stephen T. Hedetniemi,et al.  A Linear Algorithm for the Domination Number of a Tree , 1975, Inf. Process. Lett..

[8]  A. Hoffman,et al.  Totally-Balanced and Greedy Matrices , 1985 .

[9]  Martin Farber,et al.  Characterizations of strongly chordal graphs , 1983, Discret. Math..

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

[11]  Robert E. Tarjan,et al.  Algorithmic Aspects of Vertex Elimination on Graphs , 1976, SIAM J. Comput..

[12]  David S. Johnson,et al.  Computers and In stractability: A Guide to the Theory of NP-Completeness. W. H Freeman, San Fran , 1979 .

[13]  J. Mark Keil,et al.  Total Domination in Interval Graphs , 1986, Inf. Process. Lett..

[14]  Peter J. Slater,et al.  R-Domination in Graphs , 1976, J. ACM.

[15]  G. Nemhauser,et al.  Covering, Packing and Generalized Perfection , 1985 .

[16]  G. Nemhauser,et al.  The k-Domination and k-Stability Problems on Sun-Free Chordal Graphs , 1984 .

[17]  Martin Farber,et al.  Domination, independent domination, and duality in strongly chordal graphs , 1984, Discret. Appl. Math..

[18]  G. Chang.  Total domination in block graphs , 1989 .

[19]  Tohru Kikuno,et al.  A linear algorithm for the domination number of a series-parallel graph , 1983, Discret. Appl. Math..

[20]  Martin Farber,et al.  Steiner trees, connected domination and strongly chordal graphs , 1985, Networks.

[21]  Stephen T. Hedetniemi,et al.  A linear algorithm for finding a minimum dominating set in a cactus , 1986, Discret. Appl. Math..

[22]  Fanica Gavril,et al.  Algorithms for Minimum Coloring, Maximum Clique, Minimum Covering by Cliques, and Maximum Independent Set of a Chordal Graph , 1972, SIAM J. Comput..

[23]  Richard P. Anstee,et al.  Characterizations of Totally Balanced Matrices , 1984, J. Algorithms.

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