This paper discusses the forbidden state problem, as specified by generalized mutual exclusion constraints, in the context of supervisory control of discrete event systems modelled by Petri nets. The case of backward-conflict-free and free-choice uncontrollable subnets is considered and it is shown how to transform such subnets in well-formed free-choice nets. Then, the well-formed free-choice nets are decomposed in marked graph components by recurring to minimal T-invariants. The forbidden state problem is so reformulated for the obtained marked graph components into an equivalent one which is shown to be a linear programming problem. Thus, improving existing results in literature, a polynomial complexity solution, suitable for on-line control, is achieved. Free-choice relationship and cycle modelling, that frequently occur in real-life situations, are so allowed in the uncontrollable subnet.