  A pseudoline is a homeomorphic image of the real line in the plane so that
its complement is disconnected. An arrangement of pseudolines is a set of
pseudolines in which every two cross exactly once. A drawing of a graph is
pseudolinear if the edges can be extended to an arrangement of pseudolines. In
the recent study of crossing numbers, pseudolinear drawings have played an
important role as they are a natural combinatorial extension of rectilinear
drawings. A characterization of the pseudolinear drawings of $K_n$ was found
recently. We extend this characterization to all graphs, by describing the set
of minimal forbidden subdrawings for pseudolinear drawings. Our
characterization also leads to a polynomial-time algorithm to recognize
pseudolinear drawings and construct the pseudolines when it is possible.
