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**1 - 3**of**3**### On the Computational Complexity of the Domination Game

"... Abstract. The domination game is played on an arbitrary graph G by two players, Dominator and Staller. It is known that verifying whether the game domination number of a graph is bounded by a given integer k is PSPACE-complete. On the other hand, it is showed in this paper that the problem can be so ..."

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Abstract. The domination game is played on an arbitrary graph G by two players, Dominator and Staller. It is known that verifying whether the game domination number of a graph is bounded by a given integer k is PSPACE-complete. On the other hand, it is showed in this paper that the problem can be solved for a graph G in O(∆(G) · |V (G)|k) time. In the special case when k = 3 and the graph G considered has maximum diameter, the complexity is improved to O(|V (G) | · |E(G)|+ ∆(G)3).

### How long can one bluff in the domination game?

"... The domination game is played on an arbitrary graph G by two players, Domi-nator and Staller. The game is called Game 1 when Dominator starts it, and Game 2 otherwise. In this paper bluff graphs are introduced as the graphs in which every vertex is an optimal start vertex in Game 1 as well as in Gam ..."

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The domination game is played on an arbitrary graph G by two players, Domi-nator and Staller. The game is called Game 1 when Dominator starts it, and Game 2 otherwise. In this paper bluff graphs are introduced as the graphs in which every vertex is an optimal start vertex in Game 1 as well as in Game 2. It is proved that every minus graph (a graph in which Game 2 finishes faster than Game 1) is a bluff graph. A non-trivial infinite family of minus (and hence bluff) graphs is established. minus graphs with game domination number equal to 3 are char-acterized. Double bluff graphs are also introduced and it is proved that Kneser graphs K(n, 2), n ≥ 6, are double bluff. The domination game is also studied on generalized Petersen graphs and on Hamming graphs. Several generalized Petersen graphs that are bluff graphs but not vertex-transitive are found. It is proved that Hamming graphs are not double bluff.

### 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 D-game when Dominator starts it, and S-gam ..."

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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 D-game when Dominator starts it, and S-game 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 D-game when both players play optimally. Similarly, γ′g(G) is the number of moves played in S-game. 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 so-called 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 so-called double-gamburgers.