On minimum rank and zero forcing sets of a graph

For a graph G on n vertices and a field F, the minimum rank of G over F, written as mr^F (G), is the smallest possible rank over all n × n symmetric matrices over F whose (i, j)th entry (for i ≠ j) is nonzero whenever ij is an edge in G and is zero otherwise. The maximum nullity of G over F is M^F (G) = n - mr^F (G). The minimum rank problem of a graph G is to determine mr^F (G) (or equivalently, M^F (G)). This problem has received considerable attention over the years. In [F. Barioli, W. Barrett, S. Butler, S.M. Cioabǎ, D. Cvetković, S.M. Fallat, C. Godsil, W. Haemers, L. Hogben, R. Mikkelson, S. Narayan, O. Pryporova, I. Sciriha, W. So, D. Stevanović, H. van der Holst, K.V. Meulen, A.W. Wehe, AIM Minimum Rank-Special Graphs Work Group, Zero forcing sets and the minimum rank of graphs, Linear Algebra Appl. 428 (2008) 1628-1648], a new graph parameter Z (G), the zero forcing number, was introduced to bound M^F (G) from above. The authors posted an attractive question: What is the class of graphs G for which Z (G) = M^F (G) for some field F? This paper focuses on exploring the above question.