When performing an algorithm in the self-stabilizing model, a distributed system must achieve a desirable global state regardless of the initial state, whereas each node has only local information about the system. Depending on adopted assumptions concerning the model of simultaneous execution and scheduler fairness, some algorithms may differ in stabilization time or possibly not stabilize at all. Surprisingly, we show that the class of polynomially-solvable self-stabilizing problems is invariant with respect to the assumption of weak scheduler fairness. Furthermore, for systems with a single distinguished vertex we prove a much stronger equivalence, stating that synchronisation, the existence of a central scheduler and its fairness have no influence on polynomial stabilization time.