Deciding monotonicity of simple drawings of the complete graph

A drawing of a graph is {\em $x$-monotone} if every vertical line intersects each edge of the graph at most once.

We present an $O(n^5)$ time algorithm for deciding whether a simple drawing of the complete graph $K_n$ is weakly isomorphic to an $x$-monotone drawing.

We note that this algorithm can also decide whether a drawing of $K_n$ is strongly isomorphic to an $x$-monotone drawing.