Forbidden minors for the class of graphs G with ξ(G)⩽2

For a given simple graph G, S(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.], ξ(G) is defined to be the maximum corank (i.e., nullity) among A∈S(G) having the Strong Arnold Property; ξ is used to study the minimum rank/maximum eigenvalue multiplicity problem for G. Since ξ is minor monotone, the graphs G such that ξ(G)⩽k can be described by a finite set of forbidden minors. We determine the forbidden minors for ξ(G)⩽2 and present an application of this characterization to computation of minimum rank among matrices in S(G).