  We study the class of all finite directed graphs up to primitive positive
constructability. The resulting order has a unique greatest element, namely the
graph $P_1$ with one vertex and no edges. The graph $P_1$ has a unique greatest
lower bound, namely the graph $P_2$ with two vertices and one directed edge.
Our main result is a complete description of the greatest lower bounds of
$P_2$; we call these graphs submaximal. We show that every graph that is not
equivalent to $P_1$ and $P_2$ is below one of the submaximal graphs.
