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 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 (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 the game domination number gamma(g) (G) of G. Kinnersley, West, and Zamani proved that gamma(g)(G) <= 7n/11 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 gamma(g) (G) <= 5n/8 for every n-vertex isolate-free forest G. (C) 2015 Elsevier B.V. All rights reserved.