Graph Theory pp 135-148 | Cite as

My Favorite Domination Game Conjectures

  • Michael A. HenningEmail author
Part of the Problem Books in Mathematics book series (PBM)


The domination game belongs to the growing family of competitive optimization graph games, and describes a process in which two players, Dominator and Staller, with conflicting goals collaboratively produce a dominating set in an underlying host graph. The players take turns choosing a vertex from the graph, where each vertex chosen must dominate at least one vertex not dominated by the set of vertices previously chosen. The game ends when there are no more moves available. The players’ goals are completely antithetical: while Dominator wants to minimize the size of a dominating set, Staller wants to maximize it. In this chapter, we discuss some of the conjectures on domination-type game parameters.



Research supported in part by the South African National Research Foundation and the University of Johannesburg.


  1. 1.
    N. Alon, J. Balogh, B. Bollobás, T. Szabó, Game domination number. Discrete Math. 256, 23–33 (2002)MathSciNetCrossRefGoogle Scholar
  2. 2.
    B. Brešar, S. Klavžar, D.F. Rall, Domination game and an imagination strategy. SIAM J. Discrete Math. 24, 979–991 (2010)MathSciNetCrossRefGoogle Scholar
  3. 3.
    B. Brešar, S. Klavžar, D.F. Rall, Domination game played on trees and spanning subgraphs. Discrete Math. 313, 915–923 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    B. Brešar, S. Klavžar, G. Košmrlj, D.F. Rall, Domination game: extremal families of graphs for the 3/5-conjectures. Discrete Appl. Math. 161, 1308–1316 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    B. Brešar, P. Dorbec, S. Klavžar, G. Košmrlj, Domination game: effect of edge- and vertex-removal. Discrete Math. 330, 1–10 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cs. Bujtás, Domination game on forests. Discrete Math. 338, 2220–2228 (2015)Google Scholar
  7. 7.
    Cs. Bujtás, On the game domination number of graphs with given minimum degree. Electron. J. Combin. 22(3), #P3.29 (2015)Google Scholar
  8. 8.
    Cs. Bujtás, On the game total domination number. Graphs Combin. 34(3), 415–425 (2018)Google Scholar
  9. 9.
    Cs. Bujtás, M.A. Henning, Zs. Tuza, Transversal game on hypergraphs and the \(\frac {3}{4}\)-conjecture on the total domination game. SIAM J. Discrete Math. 30, 1830–1847 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    W. Goddard, M.A. Henning, The competition-independence game in trees. J. Combin. Math. Combin. Comput. 104, 161–170 (2018)MathSciNetzbMATHGoogle Scholar
  11. 11.
    T.W. Haynes, M.A. Henning, Paired-domination game played in graphs, manuscriptGoogle Scholar
  12. 12.
    T.W. Haynes, P.J. Slater, Paired-domination in graphs. Networks 32, 199–206 (1998)MathSciNetCrossRefGoogle Scholar
  13. 13.
    T.W. Haynes, P.J. Slater, Paired-domination and the paired-domatic number. Congr. Numer. 109, 65–72 (1995)MathSciNetzbMATHGoogle Scholar
  14. 14.
    M.A. Henning, W.B. Kinnersley, Domination game: a proof of the 3∕5-Conjecture for graphs with minimum degree at least two. SIAM J. Discrete Math. 30(1), 20–35 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    M.A. Henning, C. Löwenstein, Domination game: extremal families for the 3∕5-conjecture for forests. Discuss. Math. Graph Theory 37(2), 369–381 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    M.A. Henning, D.F. Rall, Progress towards the total domination game \(\frac {3}{4}\)-conjecture. Discrete Math. 339, 2620–2627(2016)Google Scholar
  17. 17.
    M.A. Henning, A. Yeo, Total Domination in Graphs. Springer Monographs in Mathematics (2013). ISBN: 978-1-4614-6524-9 (Print) 978-1-4614-6525-6 (Online)Google Scholar
  18. 18.
    M.A. Henning, S. Klavžar, D.F. Rall, Total version of the domination game. Graphs Combin. 31, 1453–1462 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    M.A. Henning, S. Klavžar, D. Rall, The 4/5 upper bound on the game total domination number. Combinatorica 37, 223–251 (2017)MathSciNetCrossRefGoogle Scholar
  20. 20.
    W.B. Kinnersley, D.B. West, R. Zamani, Extremal problems for game domination number. SIAM J. Discrete Math. 27, 2090–2107(2013)MathSciNetCrossRefGoogle Scholar
  21. 21.
    N. Marcus, D. Peleg, The domination game: proving the 3∕5 conjecture on isolate-free forests, manuscript.
  22. 22.
    J.B. Phillips, P.J. Slater, An introduction to graph competition independence and enclaveless parameters. Graph Theory Notes N. Y. 41, 37–41 (2001)MathSciNetGoogle Scholar
  23. 23.
    J.B. Phillips, P.J. Slater, Graph competition independence and enclaveless parameters. Congr. Numer. 154, 79–100 (2002)MathSciNetzbMATHGoogle Scholar
  24. 24.
    S. Schmidt, The 3∕5-conjecture for weakly S(K 1,3)-free forests. Discrete Math. 339, 2767–2774 (2016)MathSciNetCrossRefGoogle Scholar
  25. 25.
    P.J. Slater, Enclaveless sets and MK-systems. J. Res. Natl. Bur. Stand. 82, 197–202 (1977)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.University of JohannesburgDepartment of Pure and Applied MathematicsAuckland ParkSouth Africa

Personalised recommendations