Graphs and Combinatorics

, Volume 34, Issue 3, pp 415–425 | Cite as

On the Game Total Domination Number

Original Paper
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Abstract

The total domination game is a two-person competitive optimization game, where the players, Dominator and Staller, alternately select vertices of an isolate-free graph G. Each vertex chosen must strictly increase the number of vertices totally dominated. This process eventually produces a total dominating set of G. Dominator wishes to minimize the number of vertices chosen in the game, while Staller wishes to maximize it. The game total domination number of G, \(\gamma _{\mathrm{tg}}(G)\), is the number of vertices chosen when Dominator starts the game and both players play optimally. Recently, Henning, Klavžar, and Rall proved that \(\gamma _{\mathrm{tg}}(G) \le \frac{4}{5}n\) holds for every graph G which is given on n vertices such that every component of it is of order at least 3; they also conjectured that the sharp upper bound would be \(\frac{3}{4}n\). Here, we prove that \(\gamma _{\mathrm{tg}}(G) \le \frac{11}{14}n\) holds for every G which contains no isolated vertices or isolated edges.

Keywords

Dominating set Total dominating set Total domination game Open neighborhood hypergraph Transversal game 

Mathematics Subject Classification

05C69 05C65 05C57 

References

  1. 1.
    Brešar, B., Klavžar, S., Rall, D.F.: Domination game and an imagination strategy. SIAM J. Discrete Math. 24, 979–991 (2010)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Brešar, B., Dorbec, P., Klavžar, S., Košmrlj, G.: Domination game: effect of edge- and vertex-removal. Discrete Math. 330, 1–10 (2014)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    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, Veszprém, pp. 73–78 (2013)Google Scholar
  4. 4.
    Bujtás, Cs: Domination game on forests. Discrete Math. 338, 2220–2228 (2015)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bujtás, Cs.: On the game domination number of graphs with given minimum degree. Electron. J. Combin. 22, #P3.29 (2015)Google Scholar
  6. 6.
    Bujtás, Cs, Henning, M.A., Tuza, Zs: Transversal game on hypergraphs and the \(\frac{3}{4}\)-conjecture on the total domination game. SIAM J. Discrete Math. 30, 1830–1847 (2016)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bujtás, Cs, Henning, M.A., Tuza, Zs: Bounds on the game transversal number in hypergraphs. Eur. J. Combin. 59, 34–50 (2017)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Dorbec, P., Henning, M.A.: Game total domination for cycles and paths. Discrete Appl. Math. 208, 7–18 (2016)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Haynes, T.W., Hedetniemi, S.T., Slater, P.: Fundamentals of Domination in Graphs. Marcel Dekker, New York (1998)MATHGoogle Scholar
  10. 10.
    Henning, M.A., Rall, D.F.: Progress towards the total domination game \(\frac{3}{4}\)-conjecture. Discrete Math. 339, 2620–2627 (2016)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Henning, M.A., Klavžar, S., Rall, D.F.: Total version of the domination game. Graphs Combin. 31, 1453–1462 (2015)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Henning, M.A., Klavžar, S., Rall, D.F.: The 4/5 upper bound on the game total domination number. Combinatorica 31, 223–251 (2017)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kinnersley, W.B., West, D.B., Zamani, R.: Extremal problems for game domination number. SIAM J. Discrete Math. 27, 2090–2107 (2013)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Information TechnologyUniversity of PannoniaVeszprémHungary

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