The 3/5-conjecture for weakly S(K1,3)-free forests

The 3/5-conjecture for the domination game states that the game domination numbers of an isolate-free graph G on n vertices are bounded as follows: γg(G)≤3n5 and γg′(G)≤3n+25. Recent progresses have been made on the subject and the conjecture is now proved for graphs with minimum degree at least 2. One powerful tool, introduced by Bujtás, is the so-called greedy-like strategy for Dominator. In particular, using this strategy, she has proved the conjecture for isolate-free forests without leaves at distance exactly 4. In this paper, we improve this strategy to extend the result to the larger class of weakly S(K1,3)-free forests, where a weakly S(K1,3)-free forest F is an isolate-free forest without induced S(K1,3), whose leaves are leaves of F as well.