The minimum semidefinite rank of the complement of partial k-trees

For a graph G=(V,E) with V={1,…,n}, let S(G) be the set of all real symmetric n×n matrices A=[ai,j] with ai,j≠0, i≠j if and only if ij∈E. We prove the following results. If G is the complement of a partial k-tree H, then there exists a positive semidefinite matrix A∈S(G) with rank(A)≤k+2. If, in addition, k≤3 or G is k-connected, then there exist positive semidefinite matrices A∈S(G) and B∈S(H) such that rank(A)+rank(B)≤n+2.