Traditional region-based liveness-enforcing supervisors focus on (1) maximal permissiveness of not losing legal states, (2) structural simplicity of minimal number of monitors, and (3) fast computation. Lately, a number of similar approaches can achieve minimal configuration using efficient linear programming. However, it is unclear as to the relationship between the minimal configuration and the net structure. It is important to explore the structures involved for the fewest monitors required. Once the lower bound is achieved, further iteration to merge (or reduce the number of) monitors is not necessary. The minimal strongly connected resource subnet (i.e., all places are resources) that contains the set of resource places in a basic siphon is an elementary circuit. Earlier, we showed that the number of monitors required for liveness-enforcing and maximal permissiveness equals that of basic siphons for a subclass of Petri nets modelling manufacturing, called α systems. This paper extends this to systems more powerful than the α one so that the number of monitors in a minimal configuration remains to be lower bounded by that of basic siphons. This paper develops the theory behind and shows examples.