Graphs with real algebraic co-rank at most two

Recently, there have been found relations between the algebraic co-rank and the zero forcing number along with the minimum rank. We continue on this direction by giving a characterization of the graphs with real algebraic co-rank at most 2. This implies that for any graph with minimum rank at most 3, its minimum rank is bounded from above by its real algebraic co-rank. This sheds some light on the conjecture that the real minimum rank is bounded from above by the real algebraic co-rank.