  The Gram dimension $\gd(G)$ of a graph $G$ is the smallest integer $k\ge 1$
such that any partial real symmetric matrix, whose entries are specified on the
diagonal and at the off-diagonal positions corresponding to edges of $G$, can
be completed to a positive semidefinite matrix of rank at most $k$ (assuming a
positive semidefinite completion exists). For any fixed $k$ the class of graphs
satisfying $\gd(G) \le k$ is minor closed, hence it can characterized by a
finite list of forbidden minors. We show that the only minimal forbidden minor
is $K_{k+1}$ for $k\le 3$ and that there are two minimal forbidden minors:
$K_5$ and $K_{2,2,2}$ for $k=4$. We also show some close connections to
Euclidean realizations of graphs and to the graph parameter $\nu^=(G)$ of
\cite{H03}. In particular, our characterization of the graphs with $\gd(G)\le
4$ implies the forbidden minor characterization of the 3-realizable graphs of
Belk and Connelly \cite{Belk,BC} and of the graphs with $\nu^=(G) \le 4$ of van
der Holst \cite{H03}.
