A vertex ordering characterization of simple-triangle graphs

Consider two horizontal lines in the plane. A point on the top line and an interval on the bottom line define a triangle between two lines. The intersection graph of such triangles is called a simple-triangle graph. This paper shows a vertex ordering characterization of simple-triangle graphs as follows: a graph is a simple-triangle graph if and only if there is a linear ordering of the vertices that contains both an alternating orientation of the graph and a transitive orientation of the complement of the graph.