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In the domination game on the graph G, the players Dominator and Staller alternately select vertices of G. Each vertex chosen must strictly increase the number of vertices dominated. This process eventually produces a dominating set of G; Dominator aims to minimize the size of this set, while Staller aims to maximize it. The size of the dominating set produced under optimal play is the game domination number of G, denoted by γg(G).
In this paper, we prove that γg(G) ≤ 2n/3 for every n-vertex isolate-free graph G. When G has minimum degree at least 2, we prove the stronger bound γg(G) ≤ 3n/5; this resolves a special case of a conjecture due to Kinnersley, West, and Zamani [SIAM J. Discrete Math. 27 (2013), 2090–2107]. Finally, we prove that if G is an n-vertex isolate-free graph with vertices of degree 1, then γg(G) ≤ 3n/5 + [ℓ/2] + 1; in the course of establishing this result, we answer a question of Brešar, Dorbec, Klavžar, and Košmrlj [Discrete Math.330 (2014), 1–10.].
Henning, M. A. & Kinnersley, W. B. (2016). Domination Game: A proof of the 3/5-Conjecture for Graphs with Minimum Degree at Least Two. SIAM J. Discrete Math., 30(1), 20-35. doi: 10.1137/140976935
Available at: https://doi.org/10.1137/140976935
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