Graphs with a matroid number that does not exceed 2

Let $IG$ be the system of all the independent subsets of the vertices of a graph $G$. We study two classes of graphs – graphs for which $IG$ is the union of systems of independent sets of two matroids, and graphs for which $IG$ is represented in the form of an intersection of systems of independent sets of two matroids. The first of these classes is characterized in terms of prohibited generated subgraphs. For the second we prove its isomorphic completeness.