Characterisation of forests with trivial game domination numbers

Nadjafi-Arani, M. J.; Siggers, Mark; Soltani, Hossein

doi:10.1007/s10878-015-9903-9

Cited by 35 publications

(23 citation statements)

References 7 publications

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“…Other topics of the domination game that were studied include computational complexity of the game (see [1,16]), metric properties with respect to the domination game (see [3]), the domination game played on disjoint union (see [9]), realizations of the game domination numbers (see [17]), and characterizations (see [15,18]). Also, the game has motivated studies of new games such as the total version of domination game, introduced and studied in [11,12], and the disjoint domination game introduced by Bujtás and Tuza in [8].…”

Section: Introductionmentioning

confidence: 99%

Domination game on paths and cycles

Košmrlj

2017

Ars Math. Contemp.

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Domination game is a game on a simple graph played by two players, Dominator and Staller who are alternating in taking turns. In each turn a player chooses a vertex in such a way that at least one new vertex gets dominated by this move. The game ends when all vertices are dominated, and thus no legal move is possible. As the names of the players suggest, Dominator tries to finish the game as fast as possible, while Staller wants to prolong its end as long as she can. By γ g (γ g) we denote the total number of moves in the game when Dominator (resp. Staller) starts, and both players play according to their optimal strategies. In a manuscript from 2012, Kinnersley et al. determined γ g and γ g for paths and cycles, but have not yet published this very important result. In this paper we give an alternative proof for these formulas. Our approach also explicitly describes optimal strategies for both players.

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Section: Introductionmentioning

confidence: 99%

Domination game on paths and cycles

Košmrlj

2017

Ars Math. Contemp.

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“…The following result is not difficult to prove and was implicitly or explicitly (cf. [2]) used earlier and is also stated in [17].…”

Section: Preliminariesmentioning

confidence: 99%

“…The second related game, named the disjoint domination game, was studied in [9]. Among the additional investigations of the domination game we mention the complexity studies from [1], the behaviour of the game played on the disjoint union of graphs [10], and a characterization of trees T for which γ g (T ) = γ(T ) holds [17], where γ(T ) is the usual domination number of T .…”

Section: Introductionmentioning

confidence: 99%

On graphs with small game domination number

Klavžar

Košmrlj

Schmidt

2016

APPL ANAL DISCRETE M

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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 γ g (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 γ g (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 γ g (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.

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“…Following [5], the domination game has been studied further in many papers, see e.g. [3,4,6,7,15,18,24,25,26,27,28,29,30]. The notion also inspired the introduction of the total domination game on graphs [8,14,19,20,21,22,23], transversal game [9,10], disjoint domination game [12], connected domination game [2], and domination game on hypergraphs [11].…”

Section: Introductionmentioning

confidence: 99%

Fractional Domination Game

Bujtás¹,

Tuza²

2019

Electron. J. Combin.

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Given a graph $G$, a real-valued function $f: V(G) \rightarrow [0,1]$ is a fractional dominating function if $\sum_{u \in N[v]} f(u) \ge 1$ holds for every vertex $v$ and its closed neighborhood $N[v]$ in $G$. The aim is to minimize the sum $\sum_{v \in V(G)} f(v)$. A different approach to graph domination is the domination game, introduced by Brešar et al. [SIAM J. Discrete Math. 24 (2010) 979–991]. It is played on a graph $G$ by two players, namely Dominator and Staller, who take turns choosing a vertex such that at least one previously undominated vertex becomes dominated. The game is over when all vertices are dominated. Dominator wants to finish the game as soon as possible, while Staller wants to delay the end. Assuming that both players play optimally and Dominator starts, the length of the game on $G$ is uniquely determined and is called the game domination number of $G$. We introduce and study the fractional version of the domination game, where the moves are ruled by the condition of fractional domination. Here we prove a fundamental property of this new game, namely the fractional version of the so-called Continuation Principle. Moreover, we present lower and upper bounds on the fractional game domination number of paths and cycles. These estimates are tight apart from a small additive constant. We also prove that the game domination number cannot be bounded above by any linear function of the fractional game domination number.

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Characterisation of forests with trivial game domination numbers

Cited by 35 publications

References 7 publications

Domination game on paths and cycles

Domination game on paths and cycles

On graphs with small game domination number

Fractional Domination Game

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