Every triangle-free planar graph has a planar upward drawing

A planar ordered set has a triangle-free, planar covering graph; on the other hand, there are nonplanar ordered sets whose covering graphs are planar. We show thatevery triangle-free planar graph has a planar upward drawing. This planar upward drawing can be constructed in time, polynomial in the number of vertices.Our results shed light on the apparently difficult problem, of long-standing, whether there is aneffective planarity-testing procedure for an ordered set.Supported in part by the Alexander von Humboldt Stiftung.Supported in part by the Deutsche Forschungsgemeinschaft.