When quotient of a polynomial or power series ring by a monomial ideal is half-factorial

Suppose that K is a field, S=K[x1,…,xn] or S=K[[x1,…,xn]]. We present a characterization of those monomial ideals I of S, for which S/I is a half-factorial ring. This characterization is related to the structure of the base field K and also depends on the combinatorics of a graph constructed from the monomial ideal I. We also show that if R[x] or R[[x]] is half-factorial for an arbitrary commutative ring R, then R is an integral domain.