Graphs and Combinatorics

, Volume 31, Issue 5, pp 1453–1462 | Cite as

Total Version of the Domination Game

  • Michael A. Henning
  • Sandi Klavžar
  • Douglas F. Rall
Original Paper


In this paper, we continue the study of the domination game in graphs introduced by Brešar et al. (SIAM J Discret Math 24:979–991, 2010). We study the total version of the domination game and show that these two versions differ significantly. We present a key lemma, known as the Total Continuation Principle, to compare the Dominator-start total domination game and the Staller-start total domination game. Relationships between the game total domination number and the total domination number, as well as between the game total domination number and the domination number, are established.


Domination game Total domination number 

Mathematics Subject Classification

05C57 91A43 05C69 



Research supported in part by the South African National Research Foundation and the University of Johannesburg, by the Ministry of Science of Slovenia under the Grants P1-0297, and by a Grant from the Simons Foundation (#209654 to Douglas Rall) and by the Wylie Enrichment Fund of Furman University. We thank the referees for several useful suggestions.


  1. 1.
    Brešar, B., Dorbec, P., Klavžar, S., Košmrlj, G.: Domination game: effect of edge- and vertex-removal. Discret. Math. 330, 1–10 (2014)CrossRefzbMATHGoogle Scholar
  2. 2.
    Brešar, B., Klavžar, S., Rall, D.F.: Domination game and an imagination strategy. SIAM J. Discret. Math. 24, 979–991 (2010)CrossRefzbMATHGoogle Scholar
  3. 3.
    Brešar, B., Klavžar, S., Rall, D.F.: Domination game played on trees and spanning subgraphs. Discret. Math. 313, 915–923 (2013)CrossRefzbMATHGoogle Scholar
  4. 4.
    Brešar, B., Klavžar, S., Košmrlj, G., Rall, D.F.: Domination game: extremal families of graphs for the 3/5-conjectures. Discret. Appl. Math. 161, 1308–1316 (2013)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bujtás, Cs.: Domination game on trees without leaves at distance four. In: Frank, A., Recski, A., Wiener, G. (eds.) Proceedings of the 8th Japanese-Hungarian Symposium on Discrete Mathematics and Its Applications, June 4–7, pp. 73–78. Veszprém, Hungary (2013)Google Scholar
  6. 6.
    Bujtás, Cs.: Domination game on forests, arXiv:1404.1382 (2014)
  7. 7.
    Dorbec, P., Košmrlj, G., Renault, G.: The domination game played on unions of graphs. Discret. Math. (2013) (accepted)Google Scholar
  8. 8.
    Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of Domination in Graphs. Marcel Dekker Inc, New York (1998)zbMATHGoogle Scholar
  9. 9.
    Henning, M.A., Yeo, A.: Total Domination in Graphs. Springer Monographs in Mathematics. Springer, New York, ISBN-13: 978–1461465249 (2013)Google Scholar
  10. 10.
    Kinnersley, W.B., West, D.B., Zamani, R.: Extremal problems for game domination number. SIAM J. Discret. Math. 27, 2090–2107 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Kinnersley, W.B., West, D.B., Zamani, R.: Game domination for grid-like graphs (2012, unpublished manuscript)Google Scholar
  12. 12.
    Košmrlj, G.: Realizations of the game domination number. J. Comb. Optim. 28, 447–461 (2014)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Zamani, R.: Hamiltonian cycles through specified edges in bipartite graphs, domination game, and the game of revolutionaries and spies. Ph. D. Thesis, University of Illinois at Urbana-Champaign. Pro-Quest/UMI, Ann Arbor (Publication No. AAT 3496787) (2011)Google Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  • Michael A. Henning
    • 1
  • Sandi Klavžar
    • 2
    • 3
    • 4
  • Douglas F. Rall
    • 5
  1. 1.Department of MathematicsUniversity of JohannesburgJohannesburgSouth Africa
  2. 2.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia
  3. 3.Faculty of Natural Sciences and MathematicsUniversity of MariborMariborSlovenia
  4. 4.Institute of Mathematics, Physics and MechanicsLjubljanaSlovenia
  5. 5.Department of MathematicsFurman UniversityGreenvilleUSA

Personalised recommendations