# General upper bound on the game domination number

### Csilla Bujt\'as University of Pannonia, Veszpr\'em

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Minisymposium: GRAPH DOMINATION

Content: In the domination game, Dominator and Staller alternately choose a vertex of a graph $G$ and take it into a set $D$. The number of vertices dominated by the set $D$ must increase with each move and the game ends when $D$ becomes a dominating set of $G$. Dominator aims to minimize while Staller aims to maximize the number of turns. Assuming that Dominator starts and both players play optimally, the number of turns is called the game domination number $\gamma_g(G)$ of $G$. It was conjectured that the game domination number $\gamma_g(G)$ of an $n$-vertex isolate-free graph $G$ cannot be greater than $3n/5$. Here we prove that $\gamma_g(G) <0.6395 n$ always holds. This improves the earlier upper bound $2n/3$.

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