Semi-local formal fibers of minimal prime ideals of excellent reduced local rings

Let T be a complete local (Noetherian) ring containing the rationals, and let C be a finite set of incomparable non-maximal prime ideals of T partitioned into m subsets C₁,... , C_m. We find necessary and sufficient conditions for T to be the completion of an excellent reduced local ring A with precisely m minimal prime ideals J₁,.. . , J_m such that, for each i = 1, 2,... , m, the maximal elements of the set {Q ε SpecT | Q ∩ A = J_i} are precisely the elements of C_i. We also find necessary and sufficient conditions for T to be the completion of such a ring A for the case where A need not be excellent.