The incidence graph of a system of points and lines in $\mathbb{R}^2$ is the bipartite graph with partite sets $P, L$, where $P$ corresponds to the points, $L$ corresponds to the lines, and we add the edge $(p, \ell) \in P \times L$ to the graph if and only if line $\ell$ passes through point $p$ in $\mathbb{R}^2$. (We will assume that our points and lines are distinct.)
Are there any characterizations of the family of all incidence graphs of systems of points and lines (e.g., in terms of forbidden subgraphs or minors)? For example, we know that these graphs do not contain a $K_{2, 2}$ subgraph, but this is not sufficient for a complete characterization.
Alternatively, can we decide if a graph belongs to this family in polynomial time or is this problem NP-hard?