  For a standard graded Cohen-Macaulay ring $S$, if the quotient
$S/(\underline{x})$ admits non-free totally reflexive modules, where
$\underline{x}$ is a system of parameters consisting of elements of degree one,
then so does the ring $S$. As an application, we consider the question of which
Stanley-Reisner rings of graphs admit non-free totally reflexive modules.
