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Extremal problems for game domination number
, 2012
"... In the domination game on a graph G, two players called Dominator and Staller alternately select vertices of G. Each vertex chosen must strictly increase the number of vertices dominated; the game ends when the chosen set becomes a dominating set of G. Dominator aims to minimize the size of the resu ..."
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In the domination game on a graph G, two players called Dominator and Staller alternately select vertices of G. Each vertex chosen must strictly increase the number of vertices dominated; the game ends when the chosen set becomes a dominating set of G. Dominator aims to minimize the size of the resulting dominating set, while Staller aims to maximize it. When both players play optimally, the size of the dominating set produced is the game domination number of G, denoted by γg(G) when Dominator plays first and by γ ′ g(G) when Staller plays first. We prove that γg(G) ≤ 7n/11 when G is an isolatefree nvertex forest and that γg(G) ≤ ⌈7n/10 ⌉ for any isolatefree nvertex graph. In both cases we conjecture that γg(G) ≤ 3n/5 and prove it when G is a forest of nontrivial caterpillars. We also resolve conjectures of Breˇsar, Klavˇzar, and Rall by showing that always γ ′ g(G) ≤ γg(G) + 1, that for k ≥ 2 there are graphs G satisfying γg(G) = 2k and γ ′ g(G) = 2k − 1, and that γ ′ g(G) ≥ γg(G) when G is a forest. Our results follow from fundamental lemmas about the domination game that simplify its analysis and may be useful in future research. 1
Domination game played on trees and spanning subgraphs
 Discrete Mathematics
"... Domination game played on trees and spanning subgraphs ..."
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Domination game played on trees and spanning subgraphs
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 ..."
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Cited by 4 (0 self)
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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.
Domination game: effect of edge and vertexremoval
"... The domination game is played on a graph G by two players, named Dominator and Staller. They alternatively select vertices of G such that each chosen vertex enlarges the set of vertices dominated before the move on it. Dominator’s goal is that the game is finished as soon as possible, while Staller ..."
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Cited by 3 (0 self)
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The domination game is played on a graph G by two players, named Dominator and Staller. They alternatively select vertices of G such that each chosen vertex enlarges the set of vertices dominated before the move on it. Dominator’s goal is that the game is finished as soon as possible, while Staller wants the game to last as long as possible. It is assumed that both play optimally. Game 1 and Game 2 are variants of the game in which Dominator and Staller has the first move, respectively. The game domination number γg(G), and the Stallerstart game domination number γ g(G), is the number of vertices chosen in Game 1 and Game 2, respectively. It is proved that if e ∈ E(G), then γg(G) − γg(G − e)  ≤ 2 and γ g(G) − γ g(G − e)  ≤ 2, and that each of the possibilities here is realizable by connected graphs G for all values of γg(G) and γ′g(G) larger than 5. For the remaining small values it is either proved that realizations are not possible or realizing examples are provided. It is also proved that if v ∈ V (G), then γg(G) − γg(G − v) ≤ 2 and γ g(G) − γ g(G − v) ≤ 2. Possibilities here are again realizable by connected graphs G in almost all the cases, the exceptional values are treated similarly as in the edgeremoval case.
Guarded subgraphs and the domination game
, 2015
"... We introduce the concept of guarded subgraph of a graph, which as a condition lies between convex and 2isometric subgraphs and is not comparable to isometric subgraphs. Some basic metric properties of guarded subgraphs are obtained, and then this concept is applied to the domination game. In this g ..."
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We introduce the concept of guarded subgraph of a graph, which as a condition lies between convex and 2isometric subgraphs and is not comparable to isometric subgraphs. Some basic metric properties of guarded subgraphs are obtained, and then this concept is applied to the domination game. In this game two players, Dominator and Staller, alternate choosing vertices of a graph, one at a time, such that each chosen vertex enlarges the set of vertices dominated so far. The aim of Dominator is that the graph is dominated in as few steps as possible, while the aim of Staller is just the opposite. The game domination number is the number of vertices chosen when Dominator starts the game and both players play optimally. The main result of this paper is that the game domination number of a graph is not smaller than the game domination number of any guarded subgraph. Several applications of this result are presented.
Total version of the domination game
, 2014
"... In this paper, we continue the study of the domination game in graphs introduced by Brešar, Klavžar, and Rall [SIAM J. Discrete Math. 24 (2010) 979–991]. We study the total version of the domination game and show that these two versions differ significantly. We present a key lemma, known as the To ..."
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In this paper, we continue the study of the domination game in graphs introduced by Brešar, Klavžar, and Rall [SIAM J. Discrete Math. 24 (2010) 979–991]. We study the total version of the domination game and show that these two versions differ significantly. We present a key lemma, known as the Total Continuation Principle, to compare the Dominatorstart total domination game and the Stallerstart total domination game. Relationships between the game total domination number and the total domination number, as well as between the game total domination number and the domination number, are established.
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 PSPACEcomplete. 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 PSPACEcomplete. 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).