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:  ] We also prove that for a prescribed edge e of G, it is NP-hard to obtain a 0-perfect forest containing e, but one can decide if there existsa 0-perfect forest not containing e in polynomial time.