Abstract
The recently introduced total domination game is studied. This game is played on a graph G by two players, named Dominator and Staller. They alternately take turns choosing vertices of G such that each chosen vertex totally dominates at least one vertex not totally dominated by the vertices previously chosen. Dominator’s goal is to totally dominate the graph as fast as possible, and Staller wishes to delay the process as much as possible. The game total domination number, γtg (G), of G is the number of vertices chosen when Dominator starts the game and both players play optimally. The Staller-start game total domination number, γ′tg (G), of G is the number of vertices chosen when Staller starts the game and both players play optimally. In this paper it is proved that if G is a graph on n vertices in which every component contains at least three vertices, then γtg (G)≤4n/5 and γ′tg (G)≤(4n+2)/5. As a consequence of this result, we obtain upper bounds for both games played on any graph that has no isolated vertices.
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References
B. Brešar, P. Dorbec, S. Klavžar and G. Košmrlj: Domination game: effect of edge- and vertex-removal, Discrete Math. 330 (2014), 1–10.
B. Brešar, S. Klavžar and D. F. Rall: Domination game and an imagination strategy, SIAM J. Discrete Math. 24 (2010), 979–991.
B. Brešar, S. Klavžar and D. F. Rall: Domination game played on trees and spanning subgraphs, Discrete Math. 313 (2013), 915–923.
B. Brešar, S. Klavžar, G. Košmrlj and D. F. Rall: Domination game: extremal families of graphs for the 3/5-conjectures, Discrete Appl. Math. 161 (2013), 1308–1316.
Cs. Bujtás: Domination game on trees without leaves at distance four, in: Proceedings of the 8th Japanese-Hungarian Symposium on Discrete Mathematics and Its Applications (A. Frank, A. Recski, G. Wiener, eds.), June 4–7, 2013, Veszprém, Hungary, 73–78.
Cs. Bujtás: On the game domination number of graphs with given minimum degree., Electron. J. Combin. 22 (2015), #P3.29
Cs. Bujtás, S. Klavžar and G. Košmrlj: Domination game critical graphs, Discuss. Math. Graph Theory 35 (2015), 781–796.
P. Dorbec, G. Košmrlj and G. Renault: The domination game played on unions of graphs, Discrete Math. 338 (2015), 71–79.
T. W. Haynes, S. T. Hedetniemi and P. J. Slater: Fundamentals of Domination in Graphs, Marcel Dekker, Inc., New York, 1998.
M. A. Henning and 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 (2016), 20–35.
M. A. Henning, S. Klavžar and D. F. Rall: Total version of the domination game, Graphs Combin. 31 (2015), 1453–1462
M. A. Henning and A. Yeo: Total Domination in Graphs., Springer Monographs in Mathematics, (2013).
W. B. Kinnersley, D. B. West and R. Zamani: Extremal problems for game domination number, SIAM J. Discrete Math. 27 (2013), 2090–2107.
W. B. Kinnersley, D. B. West and R. Zamani: Game domination for grid-like graphs, manuscript, 2012.
G. Košmrlj: Realizations of the game domination number, J. Comb. Optim. 28 (2014), 447–461.
R. Zamani: 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).
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Henning, M.A., Klavžar, S. & Rall, D.F. The 4/5 upper bound on the game total domination number. Combinatorica 37, 223–251 (2017). https://doi.org/10.1007/s00493-015-3316-3
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DOI: https://doi.org/10.1007/s00493-015-3316-3