Advertisement

Algorithmic Aspects of Domination in Graphs

  • Gerard J. Chang
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.

Keywords

Steiner Tree Interval Graph Domination Number Chordal Graph Total Domination 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [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.zbMATHMathSciNetCrossRefGoogle Scholar
  2. [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. [3]
    K. Arvind and C. Pandu Rangan. Connected domination and Steiner set on weighted permutation graphs. Inform. Process. Lett, 41: 215–220, 1992.zbMATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    T. Asano. Dynamic programming on intervals. Internat. J. Comput. Geom. Appl, 3: 323–330, 1993.zbMATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    M. J. Atallah and S. R. Kosaraju. An efficient algorithm for maxdominance, with applications. Algorithmica, 4: 221–236, 1989.zbMATHMathSciNetCrossRefGoogle Scholar
  6. [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.zbMATHMathSciNetCrossRefGoogle Scholar
  7. [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. [8]
    H. J. Bandelt and H. M. Mulder. Distance-hereditary graphs. J. Comb. Theory, Series B, 41: 182–208, 1986.zbMATHMathSciNetCrossRefGoogle Scholar
  9. [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. [10]
    A. E. Barkauskas and L. H. Host. Finding efficient dominating sets in oriented graphs. Congr. Numer, 98: 27–32, 1993.zbMATHMathSciNetGoogle Scholar
  11. [11]
    R. E. Bellman and S. E. Dreyfus. Applied Dynamic Programming. Princeton University Press, 1962.zbMATHGoogle Scholar
  12. [12]
    P. J. Bernhard, S. T. Hedetniemi and D. P. Jacobs. Efficient sets in graphs. Discrete Appl. Math, 44: 99–108, 1993.zbMATHMathSciNetCrossRefGoogle Scholar
  13. [13]
    P. Bertolazzi and A. Sassono. A class of polynomially solvable set covering problems. SIAM J. Discrete Math, 1: 306–316, 1988.zbMATHMathSciNetCrossRefGoogle Scholar
  14. [14]
    A. A. Bertossi. Dominating sets for split and bipartite graphs. Inform. Process. Lett, 19: 37–40, 1984.zbMATHMathSciNetCrossRefGoogle Scholar
  15. [15]
    A. A. Bertossi. Total domination in interval graphs. Inform. Process. Lett, 23: 131–134, 1986.zbMATHMathSciNetCrossRefGoogle Scholar
  16. [16]
    A. A. Bertossi. On the domatic number of interval graphs. Inform. Process. Lett, 28: 275–280, 1988.zbMATHMathSciNetCrossRefGoogle Scholar
  17. [17]
    A. A. Bertossi and A. Gori. Total domination and irredundance in weighted interval graphs. SIAM J. Discrete Math, 1: 317–327, 1988.zbMATHMathSciNetCrossRefGoogle Scholar
  18. [18]
    T. A. Beyer, A. Proskurowski, S. T. Hedetniemi and S. Mitchell. Independent domination in trees. Congr. Numer, 19: 321–328, 1977.MathSciNetGoogle Scholar
  19. [19]
    N. L. Biggs, E. K. Lloyd and R. J. Wilson. Graph Theory 1736–1936. Clarendon Press, Oxford, 1986.zbMATHGoogle Scholar
  20. [20]
    M. A. Bonuccelli. Dominating sets and domatic number of circular arc graphs. Discrete Appl. Math, 12: 203–213, 1985.zbMATHMathSciNetCrossRefGoogle Scholar
  21. [21]
    K. S. Booth and J. H. Johnson. Dominating sets in chordal graphs. SIAM J. Comput, 11: 191–199, 1982.zbMATHMathSciNetCrossRefGoogle Scholar
  22. [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.MathSciNetGoogle Scholar
  23. [23]
    A. Brandstädt and H. Behrendt. Domination and the use of maximum neighbourhoods. Technical Report SM-DU-204, Univ. Duisburg, 1992.Google Scholar
  24. [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. [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. [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. [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.CrossRefGoogle Scholar
  28. [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. [29]
    A. Brandstädt and D. Kratsch. On domination problems on permutation and other graphs. Theoret. Comput. Sci, 54: 181–198, 1987.zbMATHMathSciNetCrossRefGoogle Scholar
  30. [30]
    H. Breu and D. G. Kirkpatrick. Algorithms for dominating and Steiner set problems in cocomparability. Manuscript, 1996.Google Scholar
  31. [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.zbMATHMathSciNetCrossRefGoogle Scholar
  32. [32]
    M. Burlet and J. P. Uhry. Parity graphs. Annals of Discrete Math, 21: 253–277, 1984.MathSciNetGoogle Scholar
  33. [33]
    G. J. Chang. Labeling algorithms for domination problems in sun-free chordal graphs. Discrete Appl. Math,22:21–34, 1988/89.CrossRefGoogle Scholar
  34. [34]
    G. J. Chang. Total domination in block graphs. Oper. Res. Lett, 8: 53–57, 1989.zbMATHMathSciNetCrossRefGoogle Scholar
  35. [35]
    G. J. Chang, M. Farber and Z. Tuza. Algorithmic aspects of neighborhood numbers. SIAM J. Discrete Math, 6: 24–29, 1993.zbMATHMathSciNetCrossRefGoogle Scholar
  36. [36]
    G. J. Chang and G. L. Nemhauser. R-domination of block graphs. Oper. Res. Lett, 1: 214–218, 1982.zbMATHMathSciNetCrossRefGoogle Scholar
  37. [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.zbMATHMathSciNetCrossRefGoogle Scholar
  38. [38]
    G. J. Chang and G. L. Nemhauser. Covering, packing and generalized perfection. SIAM J. Algebraic Discrete Methods, 6: 109–132, 1985.zbMATHMathSciNetCrossRefGoogle Scholar
  39. [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.zbMATHMathSciNetCrossRefGoogle Scholar
  40. [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. [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. [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. [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.zbMATHMathSciNetCrossRefGoogle Scholar
  44. [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. [45]
    M. S. Chang, S. Wu, G. J. Chang and H. G. Yeh. Domination in distance-hereditary graphs. 1996. Submitted.Google Scholar
  46. [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.zbMATHMathSciNetCrossRefGoogle Scholar
  47. [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.zbMATHCrossRefGoogle Scholar
  48. [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.zbMATHMathSciNetGoogle Scholar
  49. [49]
    E. J. Cockayne and S. T. Hedetniemi. Optimal domination in graphs. IEEE Trans. Circuits and Systems, 22: 855–857, 1975.MathSciNetCrossRefGoogle Scholar
  50. [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. [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.zbMATHMathSciNetGoogle Scholar
  52. [52]
    E. J. Cockayne and F. D. K. Roberts. Computation of dominating partitions. INFOR, 15: 94–106, 1977.zbMATHGoogle Scholar
  53. [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.zbMATHMathSciNetCrossRefGoogle Scholar
  54. [54]
    C. J. Colbourn and L. K. Stewart. Permutation graphs; connected domination and Steiner trees. Discrete Math, 86: 179–189, 1990.zbMATHMathSciNetCrossRefGoogle Scholar
  55. [55]
    D. G. Corneil. The complexity of generalized clique packing. Discrete Appl. Math, 12: 233–239, 1985.zbMATHMathSciNetCrossRefGoogle Scholar
  56. [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.zbMATHMathSciNetCrossRefGoogle Scholar
  57. [57]
    D. G. Corneil, H. Lerchs and L. Stewart. Complement reducible graphs. Discrete Appl. Math, 3: 163–174, 1981.zbMATHMathSciNetCrossRefGoogle Scholar
  58. [58]
    D. G. Corneil, S. Olariu and L. Stewart. Asteroidal triple-free graphs. SIAM J. Discrete Math. To appear.Google Scholar
  59. [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. [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.MathSciNetGoogle Scholar
  61. [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. [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.zbMATHMathSciNetCrossRefGoogle Scholar
  63. [63]
    D. G. Corneil and Y. Perl. Clustering and domination in perfect graphs. Discrete Appl. Math, 9: 27–39, 1984.zbMATHMathSciNetCrossRefGoogle Scholar
  64. [64]
    D. G. Corneil, Y. Perl and L. Stewart Burlingham. A linear recognition algorithm for cographs. SIAM J. Comput, 14: 926–934, 1985.zbMATHMathSciNetCrossRefGoogle Scholar
  65. [65]
    D. G. Corneil and L. K. Stewart. Dominating sets in perfect graphs. Discrete Math, 86: 145–164, 1990.zbMATHMathSciNetCrossRefGoogle Scholar
  66. [66]
    P. Damaschke, H. Müller and D. Kratsch. Domination in convex and chordal bipartite graphs. Inform. Process. Lett, 36: 231–236, 1990.zbMATHMathSciNetCrossRefGoogle Scholar
  67. [67]
    A. D’Atri and M. Moscarini. Distance-hereditary graphs, Steiner trees, and connected domination. SIAM J. Comput, 17: 521–538, 1988.MathSciNetCrossRefGoogle Scholar
  68. [68]
    D. P. Day, O. R. Oellermann and H. C. Swart. Steiner distance-hereditary graphs. SIAM J. Discrete Math 7: 437–442, 1994.zbMATHMathSciNetCrossRefGoogle Scholar
  69. [69]
    C. F. De Jaenisch. Applications de l’Analuse mathematique an Jen des Echecs. Petrograd, 1862.Google Scholar
  70. [70]
    F. F. Dragan. HT-graphs: centers, connected r-domination and Steiner trees. Comput. Sci. J. Moldova (Kishinev), 1: 64–83, 1993.MathSciNetGoogle Scholar
  71. [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.MathSciNetGoogle Scholar
  72. [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. [73]
    F. F. Dragan and A. Brandstädt. r-Dominating cliques in graphs with hypertree structure. Discrete Math, 162: 93–108, 1996.zbMATHMathSciNetCrossRefGoogle Scholar
  74. [74]
    S. E. Dreyfus and A. M. Law. The Art and Theory of Dynamic Programming. Academic Press, New York, 1977.zbMATHGoogle Scholar
  75. [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.zbMATHMathSciNetCrossRefGoogle Scholar
  76. [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. [77]
    S. Even, A. Pnueli and A. Lempel. Permutation graphs and transitive graphs. J. Assoc. Comput. Mach, 19 (3): 400–410, 1972.zbMATHMathSciNetCrossRefGoogle Scholar
  78. [78]
    L. Euler. Solutio problematis ad geometriam situs pertinentis. Acad. Sci. Imp. Petropol, 8: 128–140, 1736.Google Scholar
  79. [79]
    M. Farber. Domination and duality in weighted trees. Congr. Numer, 33: 3–13, 1981.MathSciNetGoogle Scholar
  80. [80]
    M. Farber. Independent domination in chordal graphs. Oper. Res. Lett, 1: 134–138, 1982.zbMATHMathSciNetCrossRefGoogle Scholar
  81. [81]
    M. Farber. Domination, independent domination and duality in strongly chordal graphs. Discrete Appl. Math, 7: 115–130, 1984.zbMATHMathSciNetCrossRefGoogle Scholar
  82. [82]
    M. Farber and J. M. Keil. Domination in permutation graphs. J. Algorithms, 6: 309–321, 1985.zbMATHMathSciNetCrossRefGoogle Scholar
  83. [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.zbMATHMathSciNetCrossRefGoogle Scholar
  84. [84]
    M. R. Fellows and M. N. Hoover. Perfect domination. Australas. J. Combin, 3: 141–150, 1991.zbMATHMathSciNetGoogle Scholar
  85. [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.zbMATHMathSciNetCrossRefGoogle Scholar
  86. [86]
    M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York, 1979.zbMATHGoogle Scholar
  87. [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.zbMATHMathSciNetCrossRefGoogle Scholar
  88. [88]
    M. C. Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York, 1980.zbMATHGoogle Scholar
  89. [89]
    M. C. Golumbic. Algorithmic aspect of perfect graphs. Annals of Discrete Math, 21: 301–323, 1984.MathSciNetGoogle Scholar
  90. [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.zbMATHMathSciNetCrossRefGoogle Scholar
  91. [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.zbMATHMathSciNetCrossRefGoogle Scholar
  92. [92]
    P. H. Hammer and F. Maffray. Completely separable graphs. Discrete Appl. Math, 27: 85–99, 1990.zbMATHMathSciNetCrossRefGoogle Scholar
  93. [93]
    E. Howorka. A characterization of distance-hereditary graphs. Quart. J. Math. Oxford Ser 2, 28: 417–420, 1977.MathSciNetCrossRefGoogle Scholar
  94. [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.MathSciNetGoogle Scholar
  95. [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. [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.MathSciNetGoogle Scholar
  97. [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.zbMATHMathSciNetGoogle Scholar
  98. [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.zbMATHMathSciNetGoogle Scholar
  99. [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. [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. [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. [102]
    S. M. Hedetniemi, S. T. Hedetniemi and D. P. Jacobs. Private domination: theory and algorithms. Congr. Numer, 79: 147–157, 1990.zbMATHMathSciNetGoogle Scholar
  103. [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. [104]
    S. T. Hedetniemi and R. C. Laskar, editors. Topics on Domination, volume 48. North Holland, New York, 1990.Google Scholar
  105. [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.zbMATHMathSciNetCrossRefGoogle Scholar
  106. [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.zbMATHMathSciNetCrossRefGoogle Scholar
  107. [107]
    W. Hsu. The distance-domination numbers of trees. Oper. Res. Lett, 1: 96–100, 1982.zbMATHCrossRefGoogle Scholar
  108. [108]
    W. Hsu and K. Tsai. Linear time algorithms on circular-arc graphs. Inform. Process. Lett, 40: 123–129, 1991.zbMATHMathSciNetCrossRefGoogle Scholar
  109. [109]
    S. F. Hwang and G. J. Chang. k-Neighbor covering and independence problem. SIAM J. Discrete Math To appear.Google Scholar
  110. [110]
    S. F. Hwang and G. J. Chang. The k-neighbor domination problem in block graphs. European J. Oper. Res, 52: 373–377, 1991.zbMATHCrossRefGoogle Scholar
  111. [111]
    S. F. Hwang and G. J. Chang. The edge domination problem. Discuss. Math.-Graph Theory, 15: 51–57, 1995.zbMATHMathSciNetGoogle Scholar
  112. [112]
    O. H. Ibarra and Q. Zheng. Some efficient algorithms for permutation graphs. J. Algorithms, 16: 453–469, 1994.zbMATHMathSciNetCrossRefGoogle Scholar
  113. [113]
    M. S. Jacobson and K. Peters. Complexity questions for n-domination and related parameters. Congr. Numer, 68: 7–22, 1989.MathSciNetGoogle Scholar
  114. [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. [115]
    D. S. Johnson. The NP-completeness column: an ongoing guide. J. Algorithms, 5: 147–160, 1984.zbMATHMathSciNetCrossRefGoogle Scholar
  116. [116]
    D. S. Johnson. The NP-completeness column: an ongoing guide. J. Algorithms, 6: 291–305, 434–451, 1985.CrossRefGoogle Scholar
  117. [117]
    J. M. Keil. Total domination in interval graphs. Inform. Process. Lett, 22: 171–174, 1986.zbMATHMathSciNetCrossRefGoogle Scholar
  118. [118]
    J. M. Keil. The complexity of domination problems in circle graphs. Discrete Appl. Math, 42: 51–63, 1993.zbMATHMathSciNetCrossRefGoogle Scholar
  119. [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.zbMATHMathSciNetCrossRefGoogle Scholar
  120. [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. [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.zbMATHMathSciNetCrossRefGoogle Scholar
  122. [122]
    E. Köhler. Connected domination on trapezoid graphs in O(n) time. Manuscript, 1996.Google Scholar
  123. [123]
    A. Kolen. Solving covering problems and the uncapacitated plant location problem on trees. Eropean J. Oper. Res, 12: 266–278, 1983zbMATHMathSciNetCrossRefGoogle Scholar
  124. [124]
    D. Kratsch. Finding dominating cliques efficiently, in strongly chordal graphs and undirected path graphs. Discrete Math, 86: 225–238, 1990.zbMATHMathSciNetCrossRefGoogle Scholar
  125. [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. [126]
    D. Kratsch, P. Damaschke and A. Lubiw. Dominating cliques in chordal graphs. Discrete Math, 128: 269–275, 1994.zbMATHMathSciNetCrossRefGoogle Scholar
  127. [127]
    D. Kratsch and L. Stewart. Domination on cocomparability graphs. SIAM J. Discrete Math, 6 (3): 400–417, 1993.zbMATHMathSciNetCrossRefGoogle Scholar
  128. [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.zbMATHMathSciNetCrossRefGoogle Scholar
  129. [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. [130]
    Y. D. Liang. Domination in trapezoid graphs. Inform. Process. Lett, 52: 309–315, 1994.MathSciNetCrossRefGoogle Scholar
  131. [131]
    Y. D. Liang. Steiner set and connected domination in trapezoid graphs. Inform. Process. Lett, 56: 101–108, 1995.MathSciNetCrossRefGoogle Scholar
  132. [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.zbMATHMathSciNetCrossRefGoogle Scholar
  133. [133]
    M. Livingston and Q. F. Stout. Constant time computation of minimum dominating sets. Congr. Numer, 105: 116–128, 1994.zbMATHMathSciNetGoogle Scholar
  134. [134]
    E. Loukakis. Two algorithms for determining a minimum independent dominating set. Internat. J. Comput. Math, 15: 213–229, 1984.zbMATHMathSciNetCrossRefGoogle Scholar
  135. [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.zbMATHMathSciNetCrossRefGoogle Scholar
  136. [136]
    K. L. Ma and C. W. H. Lam. Partition algorithm for the dominating set problem. Congr. Numer, 81: 69–80, 1991.MathSciNetGoogle Scholar
  137. [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.zbMATHMathSciNetCrossRefGoogle Scholar
  138. [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.zbMATHMathSciNetCrossRefGoogle Scholar
  139. [139]
    R. M. McConnell and J. P. Spinrad. Modular decomposition and transitive orientation. Manuscript, 1995.Google Scholar
  140. [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.zbMATHMathSciNetCrossRefGoogle Scholar
  141. [141]
    S. L. Mitchell and S. T. Hedetniemi. Edge domination in trees. Congr. Numer, 19: 489–509, 1977.MathSciNetGoogle Scholar
  142. [142]
    S. L. Mitchell and S. T. Hedetniemi. Independent domination in trees. Congr. Numer, 29: 639–656, 1979.Google Scholar
  143. [143]
    S. L. Mitchell, S. T. Hedetniemi and S. Goodman. Some linear algorithms on trees. Congr. Numer, 14: 467–483, 1975.MathSciNetGoogle Scholar
  144. [144]
    M. Moscarini. Doubly chordal graphs, Steiner trees and connected domination. Networks, 23: 59–69, 1993.zbMATHMathSciNetCrossRefGoogle Scholar
  145. [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.zbMATHMathSciNetCrossRefGoogle Scholar
  146. [146]
    K. S. Natarajan and L. J. White. Optimum domination in weighted trees. Inform. Process. Lett, 7: 261–265, 1978.zbMATHMathSciNetCrossRefGoogle Scholar
  147. [147]
    G. L. Nemhauser. Introduction to Dynamic Programming. John Wiley & Sons, 1966.Google Scholar
  148. [148]
    A. K. Parekh. Analysis of a greedy heuristic for finding small dominating sets in graphs. Inform. Process. Lett, 39: 237–240, 1991.zbMATHMathSciNetCrossRefGoogle Scholar
  149. [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. [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.zbMATHMathSciNetCrossRefGoogle Scholar
  151. [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.MathSciNetGoogle Scholar
  152. [152]
    A. Pnueli, A. Lempel and S. Even. Transitive orientation of graphs and identification of permutation graphs. Canad. J. Math, 23: 160–175, 1971.zbMATHMathSciNetCrossRefGoogle Scholar
  153. [153]
    A. Proskurowski. Minimum dominating cycles in 2-trees. Internat. J. Comput. Inform. Sci, 8: 405–417, 1979.zbMATHMathSciNetCrossRefGoogle Scholar
  154. [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. [155]
    G. Ramalingam and C. Pandu Rangan. Total domination in interval graphs revisited. Inform. Process. Lett, 27: 17–21, 1988.zbMATHMathSciNetCrossRefGoogle Scholar
  156. [156]
    G. Ramalingam and C. Pandu Rangan. A unified approach to domination problems in interval graphs. Inform. Process. Lett, 27: 271–274, 1988.zbMATHMathSciNetCrossRefGoogle Scholar
  157. [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. [158]
    J. S. Rohl. A faster lexicographic N queens algorithm. Inform. Process. Lett, 17: 231–233, 1983.CrossRefGoogle Scholar
  159. [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. [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. [161]
    P. J. Slater. R-domination in graphs. J. Assoc. Comput. Mach, 23: 446–450, 1976.zbMATHMathSciNetCrossRefGoogle Scholar
  162. [162]
    P. J. Slater. Domination and location in acyclic graphs. Networks, 17: 55–64, 1987.zbMATHMathSciNetCrossRefGoogle Scholar
  163. [163]
    C. B. Smart and P. J. Slater. Complexity results for closed neighborhood order parameters. Congr. Numer, 112: 83–96, 1995.zbMATHMathSciNetGoogle Scholar
  164. [164]
    R. Sosic and J. Gu. A polynomial time algorithm for the N-queens problem. SIGART Bull, 2 (2): 7–11, 1990.CrossRefGoogle Scholar
  165. [165]
    J. Spinrad. On comparability and permutation graphs. SIAM J. Comput, 14: 658–670, 1985.zbMATHMathSciNetCrossRefGoogle Scholar
  166. [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. [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.MathSciNetCrossRefGoogle Scholar
  168. [168]
    J. A. Telle. Complexity of domination-type problems in graphs. Nordic J. Comput, 1: 157–171, 1994.MathSciNetGoogle Scholar
  169. [169]
    J. A. Telle and A. Proskurowski. Efficient sets in partial k-trees. Discrete Appl. Math, 44: 109–117, 1993.zbMATHMathSciNetCrossRefGoogle Scholar
  170. [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. [171]
    K. H. Tsai and W. L. Hsu. Fast algorithms for the dominating set problem on permutation graphs. Algorithmica, 9: 109–117, 1993.MathSciNetCrossRefGoogle Scholar
  172. [172]
    C. Tsouros and M. Satratzemi. Tree search algorithms for the dominating vertex set problem. Internat. J. Computer Math, 47: 127–133, 1993.zbMATHCrossRefGoogle Scholar
  173. [173]
    K. White, M. Farber and W. Pulleyblank. Steiner trees, connected domination and strongly chordal graphs. Networks, 15: 109–124, 1985.zbMATHMathSciNetCrossRefGoogle Scholar
  174. [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.MathSciNetGoogle Scholar
  175. [175]
    T. V. Wimer. An O(n) algorithm for domination in k-chordal graphs. Manuscript, 1986.Google Scholar
  176. [176]
    M. Yannakakis and F. Gavril. Edge dominating sets in graphs. SIAM J. Appl. Math, 38: 264–272, 1980.MathSciNetCrossRefGoogle Scholar
  177. [177]
    H. G. Yeh and G. J. Chang. Algorithmic aspect of majority domination. Taiwanese J. Math, 1: 343–350, 1997.zbMATHMathSciNetGoogle Scholar
  178. [178]
    H. G. Yeh and G. J. Chang. Linear-time algorithms for bipartite distance-hereditary graphs. Submitted.Google Scholar
  179. [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. [180]
    C. Yen and R. C. T. Lee. The weighted perfect domination problem. Inform. Process. Lett, 35 (6): 295–299, 1990.zbMATHMathSciNetCrossRefGoogle Scholar
  181. [181]
    C. Yen and R. C. T. Lee. The weighted perfect domination problem and its variants. Discrete Appl. Math, 66: 147–160, 1996.zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Gerard J. Chang
    • 1
  1. 1.Department of Applied MathematicsNational Chiao Tung UniversityHsinchuTaiwan

Personalised recommendations