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Graphs and Combinatorics

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

On the Game Total Domination Number

  • Csilla Bujtás
Original Paper
  • 111 Downloads

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 

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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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