 In each case, the agenda properties are not only sufficient but also (if \(n \ge 3)\) necessary for the result (Nehring and Puppe 2002 [Other Internet Resources], 2010; Dokow and Holzman 2010a). Note also that path-connectedness implies non-simplicity. Therefore, non-simplicity need not be listed among the theorem’s conditions, though it is needed in the variants dropping path-connectedness.