Domination game on forests

In the domination game studied here, 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 in each single turn and the game ends when D becomes a dominating set of G. Dominator aims to minimize whilst Staller aims to maximize the number of turns (or equivalently, the size of the dominating set D obtained at the end). Assuming that Dominator starts and both players play optimally, the number of turns is called the game domination number γg(G) of G. Kinnersley, West and Zamani verified that γg(G) ≤ 7n/11 holds for every isolate-free n-vertex forest G and they conjectured that the sharp upper bound is only 3n/5. Here, we prove the 3/5-conjecture for forests in which no two leaves are at distance 4 apart. Further, we establish an upper bound γg(G) ≤ 5n/8, which is valid for every isolate-free forest G.