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The 4/5 Upper Bound on the Game Total Domination Number
"... 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 previous ..."
Abstract

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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 Stallerstart 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.
On graphs with small game domination number
"... The domination game is played on a graph G by Dominator and Staller. The two players are taking turns choosing a vertex from G such that at least one previously undominated vertex becomes dominated; the game ends when no move is possible. The game is called Dgame when Dominator starts it, and Sgam ..."
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The domination game is played on a graph G by Dominator and Staller. The two players are taking turns choosing a vertex from G such that at least one previously undominated vertex becomes dominated; the game ends when no move is possible. The game is called Dgame when Dominator starts it, and Sgame otherwise. Dominator wants to finish the game as fast as possible, while Staller wants to prolong it as much as possible. The game domination number γg(G) of G is the number of moves played in Dgame when both players play optimally. Similarly, γ′g(G) is the number of moves played in Sgame. Graphs G with γg(G) = 2, graphs with γ g(G) = 2, as well as graphs extremal with respect to the diameter among these graphs are characterized. In particular, γ′g(G) = 2 and diam(G) = 3 hold for a graph G if and only if G is a socalled gamburger. Graphs G with γg(G) = 3 and diam(G) = 6, as well as graphs G with γ′g(G) = 3 and diam(G) = 5 are also characterized. The latter can be described as the socalled doublegamburgers.