Skip to main content
SpringerLink
Log in
Book cover

Handbook of Combinatorial Optimization pp 1811–1877Cite as

Algorithmic Aspects of Domination in Graphs

  • Chapter

Abstract

Graph theory was founded by Euler [78] in 1736 as a generalization to the solution of the famous problem of the Könisberg bridges. From 1736 to 1936, the same concept as graph, but under different names, was used in various scientific fields as models of real world problems, see the historic book by Biggs, Lloyd and Wilson [19]. This chapter intents to survey the domination problem in graph theory, which is a natural model for many location problems in operations research, from an algorithmic point of view.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   259.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   329.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   329.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. Arnborg and A. Proskurowski. Linear time algorithms for NP-hard problems restricted to partial K-trees. Discrete Appl. Math, 23: 11–24, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  2. K. Arvind, H. Breu, M. S. Chang, D. G. Kirkpatrick, F. Y. Lee, Y. D. Liang, K. Madhukar, C. Pandu Rangan and A. Srinivasan. Efficient algorithms in cocomparability and trapezoid graphs. Submitted, 1996.

    Google Scholar 

  3. K. Arvind and C. Pandu Rangan. Connected domination and Steiner set on weighted permutation graphs. Inform. Process. Lett, 41: 215–220, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  4. T. Asano. Dynamic programming on intervals. Internat. J. Comput. Geom. Appl, 3: 323–330, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  5. M. J. Atallah and S. R. Kosaraju. An efficient algorithm for maxdominance, with applications. Algorithmica, 4: 221–236, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  6. M. J. Atallah, G. K. Manacher and J. Urrutia. Finding a minimum independent dominating set in a permutation graph. Discrete Appl. Math, 21: 177–183, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  7. H. Balakrishnan, A. Rajaraman and C. Pandu Rangan. Connected domination and Steiner set on asteroidal triple-free graphs. In F. Dehne, J. R. Sack, N. Santoro and S. Whitesides, editors, Proc. Workshop on Algorithms and Data Structures (WADS’93), volume 709, pages 131–141, Montreal, Canada, 1993. Springier-Verlag, Berlin.

    Google Scholar 

  8. H. J. Bandelt and H. M. Mulder. Distance-hereditary graphs. J. Comb. Theory, Series B, 41: 182–208, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  9. D. W. Bange, A. E. Barkauskas and P. J. Slater. Efficient dominating sets in graphs. In R. D. Ringeisen and F. S. Roberts, editors, Applications of Discrete Mathematics, pages 189–199. SIAM, Philadelphia, PA, 1988.

    Google Scholar 

  10. A. E. Barkauskas and L. H. Host. Finding efficient dominating sets in oriented graphs. Congr. Numer, 98: 27–32, 1993.

    MATH  MathSciNet  Google Scholar 

  11. R. E. Bellman and S. E. Dreyfus. Applied Dynamic Programming. Princeton University Press, 1962.

    MATH  Google Scholar 

  12. P. J. Bernhard, S. T. Hedetniemi and D. P. Jacobs. Efficient sets in graphs. Discrete Appl. Math, 44: 99–108, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  13. P. Bertolazzi and A. Sassono. A class of polynomially solvable set covering problems. SIAM J. Discrete Math, 1: 306–316, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  14. A. A. Bertossi. Dominating sets for split and bipartite graphs. Inform. Process. Lett, 19: 37–40, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  15. A. A. Bertossi. Total domination in interval graphs. Inform. Process. Lett, 23: 131–134, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  16. A. A. Bertossi. On the domatic number of interval graphs. Inform. Process. Lett, 28: 275–280, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  17. A. A. Bertossi and A. Gori. Total domination and irredundance in weighted interval graphs. SIAM J. Discrete Math, 1: 317–327, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  18. T. A. Beyer, A. Proskurowski, S. T. Hedetniemi and S. Mitchell. Independent domination in trees. Congr. Numer, 19: 321–328, 1977.

    MathSciNet  Google Scholar 

  19. N. L. Biggs, E. K. Lloyd and R. J. Wilson. Graph Theory 1736–1936. Clarendon Press, Oxford, 1986.

    MATH  Google Scholar 

  20. M. A. Bonuccelli. Dominating sets and domatic number of circular arc graphs. Discrete Appl. Math, 12: 203–213, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  21. K. S. Booth and J. H. Johnson. Dominating sets in chordal graphs. SIAM J. Comput, 11: 191–199, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  22. A. Brandstädt. The computational complexity of feedback vertex set, Hamiltonian circuit, dominating set, Steiner tree and bandwidth on special perfect graphs. J. Inform. Process. Cybernet, 23: 471–477, 1987.

    MathSciNet  Google Scholar 

  23. A. Brandstädt and H. Behrendt. Domination and the use of maximum neighbourhoods. Technical Report SM-DU-204, Univ. Duisburg, 1992.

    Google Scholar 

  24. A. Brandstädt, V. D. Chepoi and F. F. Dragan. Clique r-domination and clique r-packing problems on dually chordal graphs. Technical Report SM-DU-251, Univ. Duisburg, 1994.

    Google Scholar 

  25. A. Brandstädt, V. D. Chepoi and F. F. Dragan. The algorithmic use of hypertree structure and maximum neighbourhood orderings. In E. W. Mayr, G. Schmidt and G. Tinhofer, editors, Lecture Notes in Comput. Sci, 20th Internat. Workshop Graph-Theoretic Concepts in Computer Science (WG’94), volume 903, pages 65–80, Berlin, 1995. Springer-Verlag.

    Google Scholar 

  26. A. Brandstädt and F. F. Dragan. A linear-time algorithm for connected r-domination and Steiner tree on distance-hereditary graphs. Technical Report SM-DU-261, Univ. Duisburg, 1994.

    Google Scholar 

  27. A. Brandstädt, F. F. Dragan, V. D. Chepoi and V. I. Voloshin. Dually chordal graphs. In Lecture Notes in Comput. Sci., 19th Internat. Workshop Graph-Theoretic Concepts in Computer Science (WG’93),volume 790, pages 237–251, Berlin, 1993. Springer-Verlag.

    Article  Google Scholar 

  28. A. Brandstädt and D. Kratsch. On the restriction of some NP-complete graph problems to permutation graphs. In L. Budach, editor, Lecture Notes in Comput. Sci, Proc. FCT’85, volume 199, pages 53–62, Berlin, 1985. Springer-Verlag.

    Google Scholar 

  29. A. Brandstädt and D. Kratsch. On domination problems on permutation and other graphs. Theoret. Comput. Sci, 54: 181–198, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  30. H. Breu and D. G. Kirkpatrick. Algorithms for dominating and Steiner set problems in cocomparability. Manuscript, 1996.

    Google Scholar 

  31. M. W. Broin and T. J. Lowe. A dynamic programming algorithm for covering problems with (greedy) totally balanced contraint matrices. SIAM J. Algebraic and Discrete Methods, 7: 348–357, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  32. M. Burlet and J. P. Uhry. Parity graphs. Annals of Discrete Math, 21: 253–277, 1984.

    MathSciNet  Google Scholar 

  33. G. J. Chang. Labeling algorithms for domination problems in sun-free chordal graphs. Discrete Appl. Math,22:21–34, 1988/89.

    Article  Google Scholar 

  34. G. J. Chang. Total domination in block graphs. Oper. Res. Lett, 8: 53–57, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  35. G. J. Chang, M. Farber and Z. Tuza. Algorithmic aspects of neighborhood numbers. SIAM J. Discrete Math, 6: 24–29, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  36. G. J. Chang and G. L. Nemhauser. R-domination of block graphs. Oper. Res. Lett, 1: 214–218, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  37. G. J. Chang and G. L. Nemhauser. The k-domination and k-stability on sun-free chordal graphs. SIAM J. Algebraic Discrete Methods, 5: 332–345, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  38. G. J. Chang and G. L. Nemhauser. Covering, packing and generalized perfection. SIAM J. Algebraic Discrete Methods, 6: 109–132, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  39. G. J. Chang, C. Pandu Rangan and S. R. Coorg. Weighted independent perfect domination on cocomparability graphs. Discrete Appl. Math, 63: 215–222, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  40. M. S. Chang. Efficient algorithms for the domination problems on interval graphs and circular-arc graphs. In IFIP Transactions A-12, Proc. IFIP 12th World Congress, volume 1, pages 402–408, 1992.

    Google Scholar 

  41. M. S. Chang. Weighted domination on cocomparability graphs. In Lecture Notes in Comput. Sci., Proc. ISAAC’95 volume 1004, pages 121–131, Berlin, 1995. Springer-Verlag.

    Google Scholar 

  42. M. S. Chang, F. H. Hsing and S. L. Peng. Irredundance in weighted interval graphs. In Proc. National Computer Symp., pages 128–137, Taipei, Taiwan, 1993.

    Google Scholar 

  43. M. S. Chang and Y. C. Liu. Polynomial algorithms for the weighted perfect domination problems on chordal and split graphs. Inform. Process. Lett, 48: 205–210, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  44. M. S. Chang and Y. C. Liu. Polynomial algorithms for weighted perfect domination problems on interval and circular-arc graphs. J. Inform. Sci. Engineering, 10: 549–568, 1994.

    Google Scholar 

  45. M. S. Chang, S. Wu, G. J. Chang and H. G. Yeh. Domination in distance-hereditary graphs. 1996. Submitted.

    Google Scholar 

  46. G. A. Cheston, G. H. Fricke, S. T. Hedetniemi and D. P. Jacobs. On the computational complexity of upper fractional domination. Discrete Appl. Math, 27: 195–207, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  47. E. J. Cockayne, S. E. Goodman and S. T. Hedetniemi. A linear algorithm for the domination number of a tree. Inform. Process. Lett, 4: 41–44, 1975.

    Article  MATH  Google Scholar 

  48. E. J. Cockayne, B. L. Hartnell, S. T. Hedetniemi and R. Laskar. Perfect domination in graphs. J. Combin. Inform. System Sci, 18: 136–148, 1993.

    MATH  MathSciNet  Google Scholar 

  49. E. J. Cockayne and S. T. Hedetniemi. Optimal domination in graphs. IEEE Trans. Circuits and Systems, 22: 855–857, 1975.

    Article  MathSciNet  Google Scholar 

  50. E. J. Cockayne and S. T. Hedetniemi. A linear algorithm for the maximum weight of an independent set in a tree. In Proc. Seventh Southeastern Conf. on Combinatorics, Graph Theory and Computing, pages 217–228, Winnipeg, 1976. Utilitas Math.

    Google Scholar 

  51. E. J. Cockayne, G. MacGillivray and C. M. Mynhardt. A linear algorithm for 0–1 universal minimal dominating functions of trees. J. Combin. Math. Combin. Comput, 10: 23–31, 1991.

    MATH  MathSciNet  Google Scholar 

  52. E. J. Cockayne and F. D. K. Roberts. Computation of dominating partitions. INFOR, 15: 94–106, 1977.

    MATH  Google Scholar 

  53. C. J. Colbourn, J. M. Keil and L. K. Stewart. Finding minimum dominating cycles in permutation graphs. Oper. Res. Lett, 4: 13–17, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  54. C. J. Colbourn and L. K. Stewart. Permutation graphs; connected domination and Steiner trees. Discrete Math, 86: 179–189, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  55. D. G. Corneil. The complexity of generalized clique packing. Discrete Appl. Math, 12: 233–239, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  56. D. G. Corneil and J. M. Keil. A dynamic programming approach to the dominating set problem on k-trees. SIAM J. Algebraic Discrete Methods, 8: 535–543, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  57. D. G. Corneil, H. Lerchs and L. Stewart. Complement reducible graphs. Discrete Appl. Math, 3: 163–174, 1981.

    Article  MATH  MathSciNet  Google Scholar 

  58. D. G. Corneil, S. Olariu and L. Stewart. Asteroidal triple-free graphs. SIAM J. Discrete Math. To appear.

    Google Scholar 

  59. D. G. Corneil, S. Olariu and L. Stewart. Linear time algorithms for dominating pairs in asteroidal triple-free graphs. SIAM J. Comput. To appear.

    Google Scholar 

  60. D. G. Corneil, S. Olariu and L. Stewart. Computing a dominating pair in an asteroidal triple-free graph in linear time. In Proc. 4th Algorithms and Data Structures Workshop, LNCS 955, volume 955, pages 358–368. Springer, 1995.

    MathSciNet  Google Scholar 

  61. D. G. Corneil, S. Olariu and L. Stewart. A linear time algorithm to compute dominating pairs in asteroidal triple-free graphs. In Lecture Notes in Comput. Sci., Proc. 22nd Internat. Colloq. on Automata, Languages and Programming (ICALP’95) volume 994, pages 292–302, Berlin, 1995. Springer-Verlag.

    Google Scholar 

  62. D. G. Corneil, S. Olariu and L. Stewart. A linear time algorithm to compute a dominating path in an AT-free graph. Inform. Process. Lett, 54: 253–258, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  63. D. G. Corneil and Y. Perl. Clustering and domination in perfect graphs. Discrete Appl. Math, 9: 27–39, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  64. D. G. Corneil, Y. Perl and L. Stewart Burlingham. A linear recognition algorithm for cographs. SIAM J. Comput, 14: 926–934, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  65. D. G. Corneil and L. K. Stewart. Dominating sets in perfect graphs. Discrete Math, 86: 145–164, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  66. P. Damaschke, H. Müller and D. Kratsch. Domination in convex and chordal bipartite graphs. Inform. Process. Lett, 36: 231–236, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  67. A. D’Atri and M. Moscarini. Distance-hereditary graphs, Steiner trees, and connected domination. SIAM J. Comput, 17: 521–538, 1988.

    Article  MathSciNet  Google Scholar 

  68. D. P. Day, O. R. Oellermann and H. C. Swart. Steiner distance-hereditary graphs. SIAM J. Discrete Math 7: 437–442, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  69. C. F. De Jaenisch. Applications de l’Analuse mathematique an Jen des Echecs. Petrograd, 1862.

    Google Scholar 

  70. F. F. Dragan. HT-graphs: centers, connected r-domination and Steiner trees. Comput. Sci. J. Moldova (Kishinev), 1: 64–83, 1993.

    MathSciNet  Google Scholar 

  71. F. F. Dragan. Dominating cliques in distance-hereditary graphs. In Lecture Notes in Comput. Sci., Algorithm Theory-SWAT/94: 4th Scandinavian Workshop on Algorithm Theory volume 824, pages 370–381, Berlin, 1994. Springer-Verlag.

    MathSciNet  Google Scholar 

  72. F. F. Dragan and A. Brandstädt. Dominating cliques in graphs with hypertree structure. In E. M. Schmidt and S. Skyum, editors, Lecture Notes in Comput. Sci, Internat. Symp. on Theoretical Aspects of Computer Science (STACS’94), volume 775, pages 735–746, Berlin, 1994. Springer-Verlag.

    Google Scholar 

  73. F. F. Dragan and A. Brandstädt. r-Dominating cliques in graphs with hypertree structure. Discrete Math, 162: 93–108, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  74. S. E. Dreyfus and A. M. Law. The Art and Theory of Dynamic Programming. Academic Press, New York, 1977.

    MATH  Google Scholar 

  75. J. E. Dunbar, W. Goddard, S. T. Hedetniemi, M. A. Henning and A. A. McRae. The algorithmic complexity of minus domination in graphs. Discrete Appl. Math, 68: 73–84, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  76. J. E. Dunbar, S. T. Hedetniemi, M. A. Henning and A. A. McRae. Minus domination in graphs. Comput. Math. Appl. To appear.

    Google Scholar 

  77. S. Even, A. Pnueli and A. Lempel. Permutation graphs and transitive graphs. J. Assoc. Comput. Mach, 19 (3): 400–410, 1972.

    Article  MATH  MathSciNet  Google Scholar 

  78. L. Euler. Solutio problematis ad geometriam situs pertinentis. Acad. Sci. Imp. Petropol, 8: 128–140, 1736.

    Google Scholar 

  79. M. Farber. Domination and duality in weighted trees. Congr. Numer, 33: 3–13, 1981.

    MathSciNet  Google Scholar 

  80. M. Farber. Independent domination in chordal graphs. Oper. Res. Lett, 1: 134–138, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  81. M. Farber. Domination, independent domination and duality in strongly chordal graphs. Discrete Appl. Math, 7: 115–130, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  82. M. Farber and J. M. Keil. Domination in permutation graphs. J. Algorithms, 6: 309–321, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  83. A. M. Farley, S. T. Hedetniemi and A. Proskurowski. Partitioning trees: matching, domination and maximum diameter. Internat. J. Comput. Inform. Sci, 10: 55–61, 1981.

    Article  MATH  MathSciNet  Google Scholar 

  84. M. R. Fellows and M. N. Hoover. Perfect domination. Australas. J. Combin, 3: 141–150, 1991.

    MATH  MathSciNet  Google Scholar 

  85. G. H. Fricke, M. A. Henning, O. R. Oellermann and H. C. Swart. An efficient algorithm to compute the sum of two distance domination parameters. Discrete Appl. Math, 68: 85–91, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  86. M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York, 1979.

    MATH  Google Scholar 

  87. F. Gavril. Algorithms for minimum colorings, maximum clique, minimum coverings by cliques and maximum independent set of a chordal graph. SIAM J. Comput, 1: 180–187, 1972.

    Article  MATH  MathSciNet  Google Scholar 

  88. M. C. Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York, 1980.

    MATH  Google Scholar 

  89. M. C. Golumbic. Algorithmic aspect of perfect graphs. Annals of Discrete Math, 21: 301–323, 1984.

    MathSciNet  Google Scholar 

  90. D. L. Grinstead and P. J. Slater. A recurrence template for several parameters in series-parallel graphs. Discrete Appl. Math, 54: 151–168, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  91. D. L. Grinstead, P. J. Slater, N. A. Sherwani and N. D. Holmes. Efficient edge domination problems in graphs. Inform. Process. Lett, 48: 221–228, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  92. P. H. Hammer and F. Maffray. Completely separable graphs. Discrete Appl. Math, 27: 85–99, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  93. E. Howorka. A characterization of distance-hereditary graphs. Quart. J. Math. Oxford Ser 2, 28: 417–420, 1977.

    Article  MathSciNet  Google Scholar 

  94. E. O. Hare, W. R. Hare and S. T. Hedetniemi. Algorithms for computing the domination number of k × n complete grid graphs. Congr. Numer, 55: 81–92, 1986.

    MathSciNet  Google Scholar 

  95. E. O. Hare, S. Hedetniemi, R. C. Laskar, K. Peters and T. Wimer. Linear-time computability of combinatorial problems on generalizedseries-parallel graphs. In D. S. Johnson, T. Nishizeki, A. Nozaki and H. S. Wilf, editors, Discrete Algorithms and Complexity, Proc. Japan-US Joint Seminar, pages 437–457, Kyoto, Japan, 1987. Academic Press, New York.

    Google Scholar 

  96. E. O. Hare and S. T. Hedetniemi. A linear algorithm for computing the knight’s domination number of a K × N chessboard. Congr. Numer, 59: 115–130, 1987.

    MathSciNet  Google Scholar 

  97. J. H. Hattingh, M. A. Henning and P. J. Slater. On the algorithmic complexity of signed domination in graphs. Australas. J. Combin, 12, 1995. 101–112.

    MATH  MathSciNet  Google Scholar 

  98. J. H. Hattingh, M. A. Henning and J. L. Walters. On the computational complexity of upper distance fractional domination. Australas. J. Combin, 7: 133–144, 1993.

    MATH  MathSciNet  Google Scholar 

  99. T. W. Haynes, S. T. Hedetniemi and P. J. Slater, editors. Domination in Graphs: Advanced Topics. Marcel Dekker, Inc., New York, 1997.

    Google Scholar 

  100. T. W. Haynes, S. T. Hedetniemi and P. J. Slater. Fundamentals of Domination in Graphs. Marcel Dekker, Inc., New York, 1997.

    Google Scholar 

  101. S. M. Hedetniemi, S. T. Hedetniemi and M. A. Henning. The algorithmic complexity of perfect neighborhoods in graphs. J. Combin. Math. Combin. Comput To appear.

    Google Scholar 

  102. S. M. Hedetniemi, S. T. Hedetniemi and D. P. Jacobs. Private domination: theory and algorithms. Congr. Numer, 79: 147–157, 1990.

    MATH  MathSciNet  Google Scholar 

  103. S. M. Hedetniemi, S. T. Hedetniemi and R. C. Laskar. Domination in trees: models and algorithms. In Y. Alavi, G. Chartrand, L. Lesniak, D. R. Lick and C. E. Wall, editors, Graph Theory with Applications to Algorithms and Computer Science, pages 423–442. Wiley, New York, 1985.

    Google Scholar 

  104. S. T. Hedetniemi and R. C. Laskar, editors. Topics on Domination, volume 48. North Holland, New York, 1990.

    Google Scholar 

  105. S. T. Hedetniemi, R. C. Laskar and J. Pfaff. A linear algorithm for finding a minimum dominating set in a cactus. Discrete Appl. Math, 13: 287–292, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  106. A. J. Hoffman, A. W. J. Kolen and M. Sakarovitch. Totally-balanced and greedy matrices. SIAM J. Algebraic and Discrete Methods, 6: 721–730, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  107. W. Hsu. The distance-domination numbers of trees. Oper. Res. Lett, 1: 96–100, 1982.

    Article  MATH  Google Scholar 

  108. W. Hsu and K. Tsai. Linear time algorithms on circular-arc graphs. Inform. Process. Lett, 40: 123–129, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  109. S. F. Hwang and G. J. Chang. k-Neighbor covering and independence problem. SIAM J. Discrete Math To appear.

    Google Scholar 

  110. S. F. Hwang and G. J. Chang. The k-neighbor domination problem in block graphs. European J. Oper. Res, 52: 373–377, 1991.

    Article  MATH  Google Scholar 

  111. S. F. Hwang and G. J. Chang. The edge domination problem. Discuss. Math.-Graph Theory, 15: 51–57, 1995.

    MATH  MathSciNet  Google Scholar 

  112. O. H. Ibarra and Q. Zheng. Some efficient algorithms for permutation graphs. J. Algorithms, 16: 453–469, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  113. M. S. Jacobson and K. Peters. Complexity questions for n-domination and related parameters. Congr. Numer, 68: 7–22, 1989.

    MathSciNet  Google Scholar 

  114. T. S. Jayaram, G. Sri Karishna and C. Pandu Rangan. A unified approach to solving domination problems on block graphs. Report TR-TCS-90–09, Dept. of Computer Science and Eng., Indian Inst. of Technology, 1990.

    Google Scholar 

  115. D. S. Johnson. The NP-completeness column: an ongoing guide. J. Algorithms, 5: 147–160, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  116. D. S. Johnson. The NP-completeness column: an ongoing guide. J. Algorithms, 6: 291–305, 434–451, 1985.

    Article  Google Scholar 

  117. J. M. Keil. Total domination in interval graphs. Inform. Process. Lett, 22: 171–174, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  118. J. M. Keil. The complexity of domination problems in circle graphs. Discrete Appl. Math, 42: 51–63, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  119. J. M. Keil and D. Schaefer. An optimal algorithm for finding dominating cycles in circular-arc graphs. Discrete Appl. Math, 36: 25–34, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  120. T. Kikuno, N. Yoshida and Y. Kakuda. The NP-completeness of the dominating set problem in cubic planar graphs. Trans. IEEE, pages 443–444, 1980.

    Google Scholar 

  121. T. Kikuno, N. Yoshida and Y. Kakuda. A linear algorithm for the domination number of a series-parallel graph. Discrete Appl. Math, 5: 299–311, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  122. E. Köhler. Connected domination on trapezoid graphs in O(n) time. Manuscript, 1996.

    Google Scholar 

  123. A. Kolen. Solving covering problems and the uncapacitated plant location problem on trees. Eropean J. Oper. Res, 12: 266–278, 1983

    Article  MATH  MathSciNet  Google Scholar 

  124. D. Kratsch. Finding dominating cliques efficiently, in strongly chordal graphs and undirected path graphs. Discrete Math, 86: 225–238, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  125. D. Kratsch. Domination and total domination in asteroidal triple-free graphs. Technical Report Math/Inf/96/25, F.-Schiller-Univ. Jena, 1996.

    Google Scholar 

  126. D. Kratsch, P. Damaschke and A. Lubiw. Dominating cliques in chordal graphs. Discrete Math, 128: 269–275, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  127. D. Kratsch and L. Stewart. Domination on cocomparability graphs. SIAM J. Discrete Math, 6 (3): 400–417, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  128. R. C. Laskar, J. Pfaff, S. M. Hedetniemi and S. T. Hedetniemi. On the algorithmic complexity of total domination. SIAM J. Algebraic Discrete Methods, 5: 420–425, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  129. E. L. Lawler and P. J. Slater. A linear time algorithm for finding an optimal dominating subforest of a tree. In Graph Theory with Applications to Algorithms and Computer Science, pages 501–506. Wiley, New York, 1985.

    Google Scholar 

  130. Y. D. Liang. Domination in trapezoid graphs. Inform. Process. Lett, 52: 309–315, 1994.

    Article  MathSciNet  Google Scholar 

  131. Y. D. Liang. Steiner set and connected domination in trapezoid graphs. Inform. Process. Lett, 56: 101–108, 1995.

    Article  MathSciNet  Google Scholar 

  132. Y. D. Liang, C. Rhee, S. K. Dall and S. Lakshmivarahan. A new approach for the domination problem on permutation graphs. Inform. Process. Lett, 37: 219–224, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  133. M. Livingston and Q. F. Stout. Constant time computation of minimum dominating sets. Congr. Numer, 105: 116–128, 1994.

    MATH  MathSciNet  Google Scholar 

  134. E. Loukakis. Two algorithms for determining a minimum independent dominating set. Internat. J. Comput. Math, 15: 213–229, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  135. T. L. Lu, P. H. Ho and G. J. Chang. The domatic number problem in interval graphs. SIAM J. Discrete Math, 3: 531–536, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  136. K. L. Ma and C. W. H. Lam. Partition algorithm for the dominating set problem. Congr. Numer, 81: 69–80, 1991.

    MathSciNet  Google Scholar 

  137. G. K. Manacher and T. A. Mankus. Finding a domatic partition of an interval graph in time O(n). SIAM J. Discrete Math, 9: 167–172, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  138. M. V. Marathe, H. B. Hunt III and S. S. Ravi. Efficient approximation algorithms for domatic partition and on-line coloring of circular arc graphs. Discrete Appl. Math, 64: 135–149, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  139. R. M. McConnell and J. P. Spinrad. Modular decomposition and transitive orientation. Manuscript, 1995.

    Google Scholar 

  140. S. L. Mitchell, E. J. Cockayne and S. T. Hedetniemi. Linear algorithms on recursive representations of trees. J. Comput. System Sci, 18 (1): 76–85, 1979.

    Article  MATH  MathSciNet  Google Scholar 

  141. S. L. Mitchell and S. T. Hedetniemi. Edge domination in trees. Congr. Numer, 19: 489–509, 1977.

    MathSciNet  Google Scholar 

  142. S. L. Mitchell and S. T. Hedetniemi. Independent domination in trees. Congr. Numer, 29: 639–656, 1979.

    Google Scholar 

  143. S. L. Mitchell, S. T. Hedetniemi and S. Goodman. Some linear algorithms on trees. Congr. Numer, 14: 467–483, 1975.

    MathSciNet  Google Scholar 

  144. M. Moscarini. Doubly chordal graphs, Steiner trees and connected domination. Networks, 23: 59–69, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  145. H. Müller and A. Brandstädt. The NP-completeness of STEINER TREE and DOMINATING SET for chordal bipartite graphs. Theoret. Comput. Sci, 53: 257–265, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  146. K. S. Natarajan and L. J. White. Optimum domination in weighted trees. Inform. Process. Lett, 7: 261–265, 1978.

    Article  MATH  MathSciNet  Google Scholar 

  147. G. L. Nemhauser. Introduction to Dynamic Programming. John Wiley & Sons, 1966.

    Google Scholar 

  148. A. K. Parekh. Analysis of a greedy heuristic for finding small dominating sets in graphs. Inform. Process. Lett, 39: 237–240, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  149. S. L. Peng and M. S. Chang. A new approach for domatic number problem on interval graphs. Proc. National Comp. Symp. R. O. C, pages 236–241, 1991.

    Google Scholar 

  150. S. L. Peng and M. S. Chang. A simple linear time algorithm for the domatic partition problem on strongly chordal graphs. Inform. Process. Lett, 43: 297–300, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  151. J. Pfaff, R. Laskar and S. T. Hedetniemi. Linear algorithms for independent domination and total domination in series-parallel graphs. Congr. Numer, 45: 71–82, 1984.

    MathSciNet  Google Scholar 

  152. A. Pnueli, A. Lempel and S. Even. Transitive orientation of graphs and identification of permutation graphs. Canad. J. Math, 23: 160–175, 1971.

    Article  MATH  MathSciNet  Google Scholar 

  153. A. Proskurowski. Minimum dominating cycles in 2-trees. Internat. J. Comput. Inform. Sci, 8: 405–417, 1979.

    Article  MATH  MathSciNet  Google Scholar 

  154. A. Proskurowski and J. A. Telle. Algorithms for vertex partitioning problems on partial k-trees. SIAM J. Discrete Math To appear.

    Google Scholar 

  155. G. Ramalingam and C. Pandu Rangan. Total domination in interval graphs revisited. Inform. Process. Lett, 27: 17–21, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  156. G. Ramalingam and C. Pandu Rangan. A unified approach to domination problems in interval graphs. Inform. Process. Lett, 27: 271–274, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  157. C. Rhee, Y. D. Liang, S. K. Dhall and S. Lakshmivaranhan. An O(n + m) algorithm for finding a minimum-weight dominating set in a permutation graph. SIAM J. Comput, 25: 401–419, 1996.

    Google Scholar 

  158. J. S. Rohl. A faster lexicographic N queens algorithm. Inform. Process. Lett, 17: 231–233, 1983.

    Article  Google Scholar 

  159. P. Schefiier. Linear-time algorithms for NP-complete problems restricted to partial k-trees. Technical Report 03/87, IMATH, Berlin, 1987.

    Google Scholar 

  160. D. Seese. Tree-partite graphs and the complexity of algorithms. In Lecture Notes in Computer Science, FCT 85, volume 199, pages 412–421. Springer, Berlin, 1985.

    Google Scholar 

  161. P. J. Slater. R-domination in graphs. J. Assoc. Comput. Mach, 23: 446–450, 1976.

    Article  MATH  MathSciNet  Google Scholar 

  162. P. J. Slater. Domination and location in acyclic graphs. Networks, 17: 55–64, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  163. C. B. Smart and P. J. Slater. Complexity results for closed neighborhood order parameters. Congr. Numer, 112: 83–96, 1995.

    MATH  MathSciNet  Google Scholar 

  164. R. Sosic and J. Gu. A polynomial time algorithm for the N-queens problem. SIGART Bull, 2 (2): 7–11, 1990.

    Article  Google Scholar 

  165. J. Spinrad. On comparability and permutation graphs. SIAM J. Comput, 14: 658–670, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  166. A. Srinivasa Rao and C. Pandu Rangan. Linear algorithm for domatic number problem on interval graphs. Inform. Process. Lett,33:29–33, 1989/90.

    Google Scholar 

  167. A. Srinivasan Rao and C. Pandu Rangan. Efficient algorithms for the minimum weighted dominating clique problem on permutation graphs. Theoret. Comput. Sci, 91: 1–21, 1991.

    Article  MathSciNet  Google Scholar 

  168. J. A. Telle. Complexity of domination-type problems in graphs. Nordic J. Comput, 1: 157–171, 1994.

    MathSciNet  Google Scholar 

  169. J. A. Telle and A. Proskurowski. Efficient sets in partial k-trees. Discrete Appl. Math, 44: 109–117, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  170. J. A. Telle and A. Proskurowski. Practical algorithms on partial k-trees with an application to domination-type problems. In F. Dehne, J. R. Sack, N. Santoro and S. Whitesides, editors, Lecture Notes in Comput. Sci, Proc. Third Workshop on Algorithms and Data Structures (WADS’93), volume 703, pages 610–621, Montréal, 1993. Springer-Verlag.

    Google Scholar 

  171. K. H. Tsai and W. L. Hsu. Fast algorithms for the dominating set problem on permutation graphs. Algorithmica, 9: 109–117, 1993.

    Article  MathSciNet  Google Scholar 

  172. C. Tsouros and M. Satratzemi. Tree search algorithms for the dominating vertex set problem. Internat. J. Computer Math, 47: 127–133, 1993.

    Article  MATH  Google Scholar 

  173. K. White, M. Farber and W. Pulleyblank. Steiner trees, connected domination and strongly chordal graphs. Networks, 15: 109–124, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  174. T. V. Wimer. Linear algorithms for the dominating cycle problems in series-parallel graphs, partial K-trees and Hahn graphs. Congr. Numer, 56: 289–298, 1986.

    MathSciNet  Google Scholar 

  175. T. V. Wimer. An O(n) algorithm for domination in k-chordal graphs. Manuscript, 1986.

    Google Scholar 

  176. M. Yannakakis and F. Gavril. Edge dominating sets in graphs. SIAM J. Appl. Math, 38: 264–272, 1980.

    Article  MathSciNet  Google Scholar 

  177. H. G. Yeh and G. J. Chang. Algorithmic aspect of majority domination. Taiwanese J. Math, 1: 343–350, 1997.

    MATH  MathSciNet  Google Scholar 

  178. H. G. Yeh and G. J. Chang. Linear-time algorithms for bipartite distance-hereditary graphs. Submitted.

    Google Scholar 

  179. H. G. Yeh and G. J. Chang. Weighted connected domination and Steiner trees in distance-hereditary graphs. Discrete Appl. Math To appear.

    Google Scholar 

  180. C. Yen and R. C. T. Lee. The weighted perfect domination problem. Inform. Process. Lett, 35 (6): 295–299, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  181. C. Yen and R. C. T. Lee. The weighted perfect domination problem and its variants. Discrete Appl. Math, 66: 147–160, 1996.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, National Chiao Tung University, Hsinchu, 30050, Taiwan

    Gerard J. Chang

Authors
  1. Gerard J. Chang
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. University of Minnesota, Minneapolis, USA

    Ding-Zhu Du

  2. University of Florida, Gainesville, USA

    Panos M. Pardalos

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Kluwer Academic Publishers

About this chapter

Cite this chapter

Chang, G.J. (1998). Algorithmic Aspects of Domination in Graphs. In: Du, DZ., Pardalos, P.M. (eds) Handbook of Combinatorial Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0303-9_28

Download citation

  • DOI: https://doi.org/10.1007/978-1-4613-0303-9_28

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-7987-4

  • Online ISBN: 978-1-4613-0303-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation