The Gram dimension of a graph

The Gram dimension $\gd(G)$ of a graph is the smallest integer $k \ge 1$ such that, for every assignment of unit vectors to the nodes of the graph, there exists another assignment of unit vectors lying in $\oR^k$, having the same inner products on the edges of the graph. The class of graphs satisfying $\gd(G) \le k$ is minor closed for fixed $k$, so it can characterized by a finite list of forbidden minors. For $k\le 3$, the only forbidden minor is $K_{k+1}$. We show that a graph has Gram dimension at most 4 if and only if it does not have $K_5$ and $K_{2,2,2}$ as minors. We also show some close connections to the notion of $d$-realizability of graphs. In particular, our result implies the characterization of 3-realizable graphs of Belk and Connelly \cite{Belk,BC}.