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Combinatorica

, Volume 37, Issue 2, pp 223–251 | Cite as

The 4/5 upper bound on the game total domination number

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

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.

Mathematics Subject Classification (2000)

05C57 05C05 91A43 05C69 

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Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

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

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