Before revision:   ] We also prove that for a prescribed edge e of G, it is NP-hard to obtain a 0-perfect forest containing e, but we can find a 0-perfect forest not containing e in polynomial time.
After revision:  Moreover, we show that to decide whether G has a 0-perfect forest with at least |V(G)|/2+k edges, where k is the parameter, is W ] We also prove that for a prescribed edge e of G, it is NP-hard to obtain a 0-perfect forest containing e, but we can find a 0-perfect forest not containing e in polynomial time.