We study a new geometric graph parameter \egd(G), defined as the smallest
integer r≥1 for which any partial symmetric matrix which is completable to
a correlation matrix and whose entries are specified at the positions of the
edges of G, can be completed to a matrix in the convex hull of correlation
matrices of \rank at most r. This graph parameter is motivated by its
relevance to the problem of finding low rank solutions to semidefinite programs
over the elliptope, and also by its relevance to the bounded rank Grothendieck
constant. Indeed, \egd(G)\le r if and only if the rank-r Grothendieck
constant of G is equal to 1. We show that the parameter \egd(G) is minor
monotone, we identify several classes of forbidden minors for \egd(G)\le r
and we give the full characterization for the case r=2. We also show an upper
bound for \egd(G) in terms of a new tree-width-like parameter \sla(G),
defined as the smallest r for which G is a minor of the strong product of a
tree and Kr. We show that, for any 2-connected graph G=K3,3 on at
least 6 nodes, \egd(G)\le 2 if and only if \sla(G)\le 2.Comment: 33 pages, 8 Figures. In its second version, the paper has been
modified to accommodate the suggestions of the referees. Furthermore, the
title has been changed since we feel that the new title reflects more
accurately the content and the main results of the pape