For a given simple graph G, is defined to be the set of real symmetric matrices A whose (i,j)th entry is nonzero whenever i¿j and ij is an edge in G. In [F. Barioli, S. Fallat, L. Hogben, A variant on the graph parameters of Colin de Verdière: Implications to the minimum rank of graphs, Electron. J. Linear Algebra 13 (2005) 387–404.], ¿(G) is defined to be the maximum corank (i.e., nullity) among having the Strong Arnold Property; ¿ is used to study the minimum rank/maximum eigenvalue multiplicity problem for G. Since ¿ is minor monotone, the graphs G such that ¿(G)k can be described by a finite set of forbidden minors. We determine the forbidden minors for ¿(G)2 and present an application of this characterization to computation of minimum rank among matrices in .