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Domination in Chessboards

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Structures of Domination in Graphs

Part of the book series: Developments in Mathematics ((DEVM,volume 66))

Abstract

This chapter contains a survey of results that have been obtained on domination, independent domination, irredundance, and total domination in chessboard graphs, such as queens, kings, bishops, knights, and rooks graphs.

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References

  1. W. Ahrens, Mathematische Unterhaltungen und Spiele (Druck und Verlag von B. G. Teubner, Leipzig, 1910), pp. 311–312

    MATH  Google Scholar 

  2. A. Amirabadi, Values of total domination numbers and connected domination numbers in queen graph. M.S. in Computer Science, Clemson University (2005)

    Google Scholar 

  3. J. Bell, B. Stevens, A survey of known results and research areas for n queens. Discrete Math. 309, 1–31 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. R. Berghammer, Relation-algebraic modeling and solution of chessboard independence and domination problems. J. Logic Algebraic Program. 81, 625–642 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Bezzel, Schachfreund. Berliner Schachzeitung 3, 363 (1848)

    Google Scholar 

  6. W. Bird, Computational methods for domination problems. Ph.D. Thesis, Dept. Computer Science, University of Victoria (2017)

    Google Scholar 

  7. J.-P. Bode, H. Harborth, Independent chess pieces on Euclidean boards. J. Combin. Math. Combin. Comput. 33, 209–223 (2000)

    MathSciNet  MATH  Google Scholar 

  8. J.-P. Bode, H. Harborth, Independence for knights on hexagon and triangle boards. Discrete Math. 272, 27–35 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. J.-P. Bode, H. Harborth, M. Harborth, King independence on triangle boards. Discrete Math. 266(1–3), 101–107 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. T.M. Brown, Non-attacking arrangements of n queens with initial placements. J. Recreat. Math. 38(1), 23–37 (2014)

    MathSciNet  Google Scholar 

  11. P.A. Burchett, Paired, total, and connected domination on the queen’s graph. Congr. Numer. 178, 207–222 (2006)

    MathSciNet  MATH  Google Scholar 

  12. P.A. Burchett, On the border queens problem and k-tuple domination on the rook’s graph. Congr. Numer. 209, 179–187 (2011)

    MathSciNet  MATH  Google Scholar 

  13. P.A. Burchett, An addendum regarding double domination on the queen’s graph. Congr. Numer. 209, 197 (2011)

    MathSciNet  MATH  Google Scholar 

  14. P.A. Burchett, k-tuple domination on the bishop’s graph. Util. Math. 101, 351–358 (2016)

    Google Scholar 

  15. P.A. Burchett, Paired, total and connected domination on the queen’s graph revisited. Open J. Discrete Math. 6, 1–6 (2016)

    Article  Google Scholar 

  16. P.A. Burchett, J. Buckley, The kth-border bishop domination problem. Util. Math. 100, 3–21 (2016)

    MathSciNet  MATH  Google Scholar 

  17. P.A. Burchett, R.D. Chatham, Some results for chessboard separation problems. AKCE Int. J. Graphs Comb. 15(3), 251–260 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  18. P.A. Burchett, D. Lane, J.A. Lachniet, k-tuple and k-domination on the rook’s graph and other results. Congr. Numer. 199, 187–204 (2009)

    Google Scholar 

  19. A.P. Burger, Domination in the Queens graph. Master’s Thesis, University of South Africa (1994)

    Google Scholar 

  20. A.P. Burger, The queens domination problem. Doctoral Thesis, University of South Africa (1998)

    Google Scholar 

  21. A.P. Burger, C.M. Mynhardt, Queens on hexagonal boards. Papers in honour of Stephen T. Hedetniemi. J. Combin. Math. Combin. Comput. 31, 97–111 (1999)

    MathSciNet  MATH  Google Scholar 

  22. A.P. Burger, C.M. Mynhardt, Small irredundance numbers for queens graphs. Papers in honour of Ernest J. Cockayne. J. Combin. Math. Combin. Comput. 33, 33–43 (2000)

    MathSciNet  MATH  Google Scholar 

  23. A.P. Burger, C.M. Mynhardt, Properties of dominating sets of the queens graph Q 4k+1. Util. Math. 57, 237–253 (2000)

    MathSciNet  MATH  Google Scholar 

  24. A.P. Burger, C.M. Mynhardt, An upper bound for the minimum number of queens covering the n × n chessboard. Discrete Appl. Math. 121(1–3), 51–60 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  25. A.P. Burger, C.M. Mynhardt, An improved upper bound for queens domination numbers. Discrete Math. 266, 119–131 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  26. A.P. Burger, C.M. Mynhardt, E.J. Cockayne, Domination numbers for the queen’s graph. Bull. Inst. Combin. Appl. 10, 73–82 (1994)

    MathSciNet  MATH  Google Scholar 

  27. A.P. Burger, E.J. Cockayne, C.M. Mynhardt, Domination and irredundance in the queen’s graph. Discrete Math. 163, 47–66 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  28. A.P. Burger, C.M. Mynhardt, E.J. Cockayne, Queens graphs for chessboards on the torus. Australas. J. Combin. 24, 231–246 (2001)

    MathSciNet  MATH  Google Scholar 

  29. A.P. Burger, C.M. Mynhardt, W.D. Weakley, The domination number of the toroidal queens graph of size 3k × 3k. Australas. J. Combin. 28, 137–148 (2003)

    Google Scholar 

  30. P.J. Campbell, Gauss and the eight queens problem: a study in miniature of the propagation of historical error. Hist. Math. 4, 397–404 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  31. D. Chatham, Independence and domination on shogiboard graphs. Recreat. Math. Mag. (8), 25–37 (2018)

    MathSciNet  Google Scholar 

  32. R.D. Chatham, M. Doyle, G.H. Fricke, J. Reitmann, R.D. Skaggs, M. Wolff, Independence and domination separation on chessboard graphs. J. Combin. Math. Combin. Comput. 68, 3–17 (2009)

    MathSciNet  MATH  Google Scholar 

  33. H.-C. Chen, T.-Y. Ho, The rook problem on saw-toothed chessboards. Appl. Math. Lett. 21, 1234–1237 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. E.J. Cockayne, Chessboard domination problems. Discrete Math. 86(1–3), 13–20 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  35. E. Cockayne, S. Hedetniemi, On the diagonal queens domination problem. J. Combin. Theory Ser. B 42, 137–139 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  36. E.J. Cockayne, C.M. Mynhardt, Properties of queens graphs and the irredundance number of Q 7. Aust. J. Commun. 23, 285–299 (2001)

    MathSciNet  MATH  Google Scholar 

  37. E.J. Cockayne, P.H. Spencer, On the independent queens covering problem. Graphs Combin. 4(2), 101–110 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  38. E.J. Cockayne, R.M. Dawes, S.T. Hedetniemi, Total domination in graphs. Networks 10, 211–219 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  39. E.J. Cockayne, O. Favaron, C. Payan, A.G. Thomason, Contributions to the theory of domination, independence and irredundance in graphs. Discrete Math. 33, 249–258 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  40. E. Cockayne, B. Gamble, B. Shepherd, Domination of chessboards by queens in a column. Ars Combin. 19, 105–118 (1985)

    MathSciNet  MATH  Google Scholar 

  41. E.J. Cockayne, B. Gamble, B. Shepherd, Domination parameters for the bishop’s graph. Discrete Math. 58, 221–227 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  42. A.S. Cooper, O. Pikhurko, J.R. Schmitt, G.S. Warrington, Martin Gardner’s minimum no-3-in-a-line problem. Amer. Math. Monthly 121(3), 213–221 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  43. C.F. de Jaenisch, Applications de l’Analyse Mathematique au Jeu des Echecs. Petrograd (1862)

    Google Scholar 

  44. J. DeMaio, A. Lightcap, King’s total domination number on the square of side n. Ars Combin. 113, 139–150 (2014)

    Google Scholar 

  45. J. DeMaio, H.L. Tran, Domination and independence on a triangular honeycomb chessboard. College Math. J. 44, 307–314 (2013)

    Article  MathSciNet  Google Scholar 

  46. H.E. Dudeney, The Canterbury Puzzles and Other Curious Problems (E. P. Dutton and Company, New York, 1908)

    Google Scholar 

  47. M. Eisenstein, C. Grinstead, B. Hahne, D. Van Stone, The queen domination problem. Congr. Numer. 91, 189–193 (1992)

    MathSciNet  MATH  Google Scholar 

  48. O. Favaron, G.H. Fricke, D. Pritikin, J. Puech, Irredundance and domination in kings graphs. Discrete Math. 262(1–3), 131–147 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  49. H. Fernau, Minimum Dominating Set of Queens: a trivial programming exercise? Discrete Appl. Math. 158(4), 308–318 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  50. D. Finozhenok, W.D. Weakley, An improved lower bound for domination numbers of the queen’s graph. Australas. J. Combin. 37, 295–300 (2007)

    MathSciNet  MATH  Google Scholar 

  51. D.C. Fisher, P. Thalos, Bishop covers of rectangular boards. Ars Combin. 65, 199–208 (2002)

    MathSciNet  MATH  Google Scholar 

  52. G.H. Fricke, S.M. Hedetniemi, S.T. Hedetniemi, A.A. McRae, C.K. Wallis, M.S. Jacobson, H.W. Martin, W.D. Weakley, Combinatorial problems on chessboards: a brief survey, in Graph Theory, Combinatorics, and Algorithms (Kalamazoo,1992), vols. 1, 2 (Wiley-Intersci. Publ., Wiley, New York, 1995), pp. 507–528

    Google Scholar 

  53. M. Gardner, Mathematical games – problems that are built on the knight’s move in chess. Sci. Am. 128–132 (1967)

    Google Scholar 

  54. M. Gardner, Mathematical games. Sci. Am. 125–129 (1967)

    Google Scholar 

  55. M. Gardner, Mathematical games. Sci. Am. 124–127 (1968)

    Google Scholar 

  56. D.K. Garnick, N.A. Nieuwejaar, Total domination of the m × n chessboard by kings, crosses, and knights. Ars Combin. 41, 65–75 (1995)

    MathSciNet  MATH  Google Scholar 

  57. I.P. Gent, C. Jefferson, P. Nightingale, Complexity of n-queens completion. J. Artif. Intell. Res. 59, 815–848 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  58. P.B. Gibbons, J.A. Webb, Some new results for the queens domination problem. Aust. J. Commun. 15, 145–160 (1997)

    MathSciNet  MATH  Google Scholar 

  59. D. Goncalves, A. Pinlou, M. Rao, S. Thomasse, The domination number of grids. SIAM J. Disc. Math. 25(3), 1443–1453 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  60. C.M. Grinstead, B. Hahne, D. Van Stone, On the queen domination problem. Discrete Math. 86(1–3), 21–26 (1991)

    MathSciNet  MATH  Google Scholar 

  61. R.K. Guy, Unsolved Problems in Number Theory, vol. I (Springer, Berlin, 1981)

    Book  MATH  Google Scholar 

  62. E.O. Hare, 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  MATH  Google Scholar 

  63. S. Hedetniemi, S. Hedetniemi, R. Reynolds, Combinatorial problems on chessboards: II, in Domination in Graphs: Advanced Topics, ed. by T. Haynes, S. Hedetniemi, P. Slater (Marcel Dekker, New York, 1998), pp. 507–528

    Google Scholar 

  64. M.D. Kearse, P.B. Gibbons, Computational methods and new results for chessboard problems. Aust. J. Commun. 23, 253–284 (2001)

    MathSciNet  MATH  Google Scholar 

  65. M.D. Kearse, P.B. Gibbons, A new lower bound on upper irredundance in the queen’s graph. Discrete Math. 256(1–2), 225–242 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  66. E.G. Koehler, Domination problems in three-dimensional chessboard graphs. M.S. Computer Science, Clemson University (1994)

    Google Scholar 

  67. D. König, Theorie der Endlichen und Unendlichen Graphen (Akademische VerlagsgesellschaftM. B. H., Leipzig, 1936), later Chelsea, New York, 1950

    Google Scholar 

  68. L. Larson, A theorem about primes proved on a chessboard. Math. Mag. 50(2), 69–74 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  69. B. Lemaire, Covering the 11 × 11 chessboard with knights. J. Recreat. Math. 6(4), 292 (1973)

    Google Scholar 

  70. R.M. Low, A. Kapbasov, Non-attacking bishop and king positions on regular and cylindrical chessboards. J. Integer. Seq. 20(6), 26 (2017). Art. 17.6.1

    Google Scholar 

  71. B. Messick, Finding irredundant sets in queen’s graphs using parallel genetic algorithms. M.S. in Computer Science, Clemson University (1995)

    Google Scholar 

  72. C.M. Mynhardt, Upper bounds for the domination numbers of toroidal queens graphs. Discuss. Math. Graph Theory 23(1), 163–175 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  73. F. Nauck, Briefwechsel mit allen für alle. Illustrirte Zeitung 15, 182 (1850)

    Google Scholar 

  74. P.R. Östergard, W.D. Weakley, Values of domination numbers of the queen’s graph. Electron. J. Combin. 8(1), 19 (2001). Research Paper 29

    Google Scholar 

  75. B. Pahlavsay, E. Palezzato, M. Torielli, 3-tuple total domination number of rook’s graphs. Preprint (2018)

    Google Scholar 

  76. E. Pauls, Das Maximalproblem der Damen auf dem Schachbrett. Dtsch. Schwesternztg. Organ für das Gesammte Schachleben 29(5), 129–134 (1874)

    Google Scholar 

  77. E. Pauls, Das Maximalproblem der Damen auf dem Schachbrete, II. Dtsch. Schwesternztg. Organ für das Gesammte Schachleben 29(9), 257–267 (1874)

    Google Scholar 

  78. V. Raghavan, S.M. Venkatesan, On bounds for a covering problem. Inform. Process. Lett. 25, 281–284 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  79. W.W. Rouse Ball, H.S.M. Coxeter, Mathematical Recreations and Essays, 13th edn. (Dover, Illinois, 1987)

    Google Scholar 

  80. W.W. Rouse Ball, Mathematical Recreations and Essays, 4th edn. (Macmillan, New York, 1905)

    MATH  Google Scholar 

  81. W.W. Rouse Ball, Mathematical Recreations and Essays, revision by H.S.M. Coxeter of the original 1892 Edition, Chapter 6, Minimum Pieces Problem, 3rd edn. (Macmillan, London, 1939)

    Google Scholar 

  82. A. Sinko, P. Slater, An introduction to influence parameters for chessboard graphs. Congr. Numer. 172, 15–27 (2005)

    MathSciNet  MATH  Google Scholar 

  83. A. Sinko, P.J. Slater, Efficient domination in knights graphs. AKCE Int. J. Graphs Comb. 3(2), 193–204 (2006)

    MathSciNet  MATH  Google Scholar 

  84. A. Sinko, P.J. Slater, Queen’s domination using border squares and (A, B)-restricted domination. Discrete Math. 308(20), 4822–4828 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  85. B. Steinbach, C. Posthoff, New results based on Boolean models, in Proceedings of the 9th International Workshop on Boolean Problems, ed. by B. Steinbach (TU Bergakademie, Freiberg, 2010), pp. 29–36

    Google Scholar 

  86. W.F.D. Theron, A.P. Burger, Queen domination of hexagonal hives. J. Combin. Math. Combin. Comput. 32, 161–172 (2000)

    MathSciNet  MATH  Google Scholar 

  87. W.F.D. Theron, G. Geldenhuys, Domination by queens on a square beehive. Discrete Math. 178(1–3), 213–220 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  88. R. Wagner, R. Geist, The crippled queen placement problem. Sci. Comput. Program. 4, 221–248 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  89. C.K. Wallis, Domination parameters of line graphs, of designs and variations of chessboard graphs. Ph.D. Thesis, Dept. Mathematical Sciences, Clemson University (1994)

    Google Scholar 

  90. J.J. Watkins, Across the Board: The Mathematics of Chessboard Problems (Princeton University Press, Princeton, 2004)

    Book  MATH  Google Scholar 

  91. J.J. Watkins, C. Ricci, Kings domination on a torus, in The Ninth Quadrennial International Conference on Graph Theory, Combinatorics, Algorithms and Applications. Electronic Notes in Discrete Mathematics, vol. 11 (Elsevier Sci. B. V., Amsterdam, 2002), 6 pp.

    Google Scholar 

  92. W.D. Weakley, Domination in the queen’s graph, in Graph Theory, Combinatorics, and Algorithms (Kalamazoo, 1992), vols. 1, 2 (Wiley-Intersci. Publ., Wiley, New York, 1995), pp. 1223–1232

    Google Scholar 

  93. W.D. Weakley, Upper bounds for domination numbers of the queen’s graph. Discrete Math. 242(1–3), 229–243 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  94. W.D. Weakley, Erratum to: “Upper bounds for domination numbers of the queen’s graph” [Discrete Math. 242(1–3), 229–243 (2002)]. Discrete Math. 277(1–3), 321 (2004)

    Google Scholar 

  95. W.D. Weakley, Queens around the world in twenty-five years, in Graph Theory, Favorite Conjectures and Open Problems, ed. by R. Gera, T.W. Haynes, S.T. Hedetniemi (Springer, Berlin, 2018)

    Google Scholar 

  96. A.M. Yaglom, I.M. Yaglom, Challenging Mathematical Problems with Elementary Solutions, Combinatorial Analysis and Probability Theory, vol. I (Holden-Day Inc., San Francisco, 1964)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Wilkes Honors College, Florida Atlantic University, Jupiter, FL, 33458, USA

    Jason T. Hedetniemi

  2. School of Computing, Clemson University, Clemson, SC, USA

    Stephen T. Hedetniemi

Authors
  1. Jason T. Hedetniemi
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  2. Stephen T. Hedetniemi
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Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, East Tennessee State University, Department of Mathematics and Applied Mathematics, University of Johannesburg, Johannesburg, South Africa, Johnson City, TN, USA

    Teresa W. Haynes

  2. School of Computing, Clemson University, Clemson, SC, USA

    Stephen T. Hedetniemi

  3. Department of Mathematics and Applied Mathematics, University of Johannesburg, Johannesburg, South Africa

    Michael A. Henning

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Hedetniemi, J.T., Hedetniemi, S.T. (2021). Domination in Chessboards. In: Haynes, T.W., Hedetniemi, S.T., Henning, M.A. (eds) Structures of Domination in Graphs . Developments in Mathematics, vol 66. Springer, Cham. https://doi.org/10.1007/978-3-030-58892-2_12

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