Total Version of the Domination Game

Henning, Michael A.; Klavžar, Sandi; Rall, Douglas F.

doi:10.1007/s00373-014-1470-9

Cited by 67 publications

(66 citation statements)

References 8 publications

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“…In this paper, we continue the study of the total domination game which was first investigated in [19]. A vertex totally dominates another vertex if they are neighbors.…”

Section: Introductionmentioning

confidence: 92%

“…In [19], the authors prove a Total Continuation Principle lemma from which one can readily deduce that |γ tg (G) − γ tg (G)| ≤ 1 for every graph G with no isolated vertex. Determining the exact value of γ tg (G) and γ tg (G) is a challenging problem, and is currently only known for paths and cycles [12].…”

Section: Introductionmentioning

confidence: 99%

“…[15]) and independently by Bodlaender [2] for general graphs; it has seen extensive study, see the survey [28]. Recently, work has been done on competitive optimization variants of list-colouring [5] and its more studied related version called paintability as introduced in [26] (for further references see Section 8 of [28]), matching [11], domination [4], total domination [19], disjoint domination [8], Ramsey theory [9,16,17], and more [3].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Transversal Game on Hypergraphs and the $\frac{3}{4}$-Conjecture on the Total Domination Game

Bujtás¹,

Henning²,

Tuza³

2016

SIAM J. Discrete Math.

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The 3 4 -Game Total Domination Conjecture posed by Henning, Klavžar and Rall [Combinatorica, to appear] states that if G is a graph on n vertices in which every component contains at least three vertices, then γ tg (G) ≤ 3 4 n, where γ tg (G) denotes the game total domination number of G. Motivated by this conjecture, we raise the problem to a higher level by introducing a transversal game in hypergraphs. We define the game transversal number, τ g (H), of a hypergraph H, and prove that if every edge of H has size at least 2, and H C 4 , then τ g (H) ≤ 4 11 (n H + m H ), where n H and m H denote the number of vertices and edges, respectively, in H. Further, we characterize the hypergraphs achieving equality in this bound. As an application of this result, we prove that if G is a graph on n vertices with minimum degree at least 2, then γ tg (G) < 8 11 n. As a consequence of this result, the 3 4 -Game Total Domination Conjecture is true over the class of graphs with minimum degree at least 2.

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“…In this paper, we continue the study of the total domination game which was first investigated in [19]. A vertex totally dominates another vertex if they are neighbors.…”

Section: Introductionmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Transversal Game on Hypergraphs and the $\frac{3}{4}$-Conjecture on the Total Domination Game

Bujtás¹,

Henning²,

Tuza³

2016

SIAM J. Discrete Math.

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“…The total version of the domination game was investigated in [12] and further studied in [13] where it was proved that for any graph of order n in which every component contains at least three vertices, the corresponding total invariant is bounded above by 4n/5. The second related game, named the disjoint domination game, was studied in [9].…”

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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“…A is a dominating set of G if it dominates G. The domination number γ (G) of G is the minimum size of a dominating set. There are numerous papers about this well known graph parameter; we restrict our attention to the so called 'Domination Game', introduced recently in Brešar et al (2010) and since elaborated on in Brešar et al (2014), Brešar et al (2015), Brešar et al (2013), Dorbec et al (2015), Henning et al (2014), Kinnersley et al (2013), and Košmrlj (2014).…”

Section: Introductionmentioning

confidence: 99%

Characterisation of forests with trivial game domination numbers

Nadjafi-Arani¹,

Siggers

Soltani

2015

J Comb Optim

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In the domination game, two players, the Dominator and Staller, take turns adding vertices of a fixed graph to a set, at each turn increasing the number of vertices dominated by the set, until the final set A * dominates the whole graph. The Dominator plays to minimise the size of the set A * while the Staller plays to maximise it. A graph is D-trivial if when the Dominator plays first and both players play optimally, the set A * is a minimum dominating set of the graph. A graph is S-trivial if the same is true when the Staller plays first. We consider the problem of characterising D-trivial and S-trivial graphs. We give complete characterisations of D-trivial forests and of S-trivial forests. We also show that 2-connected D-trivial graphs cannot have large girth, and conjecture that the same holds without the connectivity condition.

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Total Version of the Domination Game

Cited by 67 publications

References 8 publications

Transversal Game on Hypergraphs and the $\frac{3}{4}$-Conjecture on the Total Domination Game

Transversal Game on Hypergraphs and the $\frac{3}{4}$-Conjecture on the Total Domination Game

On graphs with small game domination number

Characterisation of forests with trivial game domination numbers

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