On minimal rank over finite fields

Let F be a field. Given a simple graph G on n vertices, its minimal rank ( with respect to F) is the minimum rank of a symmetric n x n F-valued matrix whose off-diagonal zeroes are the same as in the adjacency matrix of G. If F is finite, then for every k, it is shown that the set of graphs of minimal rank at most k is characterized by finitely many forbidden induced subgraphs, each on at most (|F|(k)/2 + 1)(2) vertices. These findings also hold in a more general context.