Forbidden minor characterizations for low-rank optimal solutions to semidefinite programs over the elliptope

•We introduce a new graph parameter related to bounded rank positive semidefinite matrix completions.•We introduce a new tree-width-like graph parameter related to the “largeur d'arborescence” defined by Colin de Verdiere.•We identify the full list of forbidden minors for both parameters for values at most 2.•We identify three families of forbidden minors for all values.•We establish links with the bounded rank Grothendieck constant.

We study a new geometric graph parameter egd(G)egd(G), defined as the smallest integer r⩾1r⩾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)⩽regd(G)⩽r if and only if the rank-r Grothendieck constant of G   is equal to 1. We show that the parameter egd(G)egd(G) is minor monotone, we identify several classes of forbidden minors for egd(G)⩽regd(G)⩽r and we give the full characterization for the case r=2r=2. We also show an upper bound for egd(G)egd(G) in terms of a new tree-width-like parameter la⊠(G)la⊠(G), defined as the smallest r for which G   is a minor of the strong product of a tree and KrKr. We show that, for any 2-connected graph G≠K3,3G≠K3,3 on at least 6 nodes, egd(G)⩽2egd(G)⩽2 if and only if la⊠(G)⩽2la⊠(G)⩽2.