An almost embedding of a graph $G$ is a drawing of $G$ where only adjacent edges may cross. A drawing of a graph is reduced if it contains no empty bigon; equivalently, the number of crossings in the drawing cannot be decreased by homotopies of the edges that do not pass over the vertices. We characterize the class of graphs that have a reduced almost embedding on the sphere with at least one crossing. As an application we improve and generalize a recent result of Garaev about $K_5$ minus an edge answering a question of A. Skopenkov and Karasev.
Joint work with J. Kynčl.