General upper bound on the game domination number

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It is conjectured that the game domination number is at most 3n∕5 for every n-vertex graph which does not contain isolated vertices. It was proved in the recent years that the conjecture holds for several graph classes, including the class of forests and that of graphs with minimum degree at least two. Here we prove that the slightly bigger upper bound 5n∕8 is valid for every isolate-free graph.