[1] These graphs are closely related to partially ordered sets and lattices.
The Hasse diagram of a partially ordered set is a directed acyclic graph whose vertices are the set elements, with an edge from x to y for each pair x, y of elements for which x ≤ y in the partial order but for which there does not exist z with x ≤ y ≤ z.
If the vertices of an st-planar graph are partially ordered by reachability, then this ordering always forms a two-dimensional complete lattice, whose Hasse diagram is the transitive reduction of the given graph.
Conversely, the Hasse diagram of every two-dimensional complete lattice is always an st-planar graph.
[2] Based on this two-dimensional partial order property, every st-planar graph can be given a dominance drawing, in which for every two vertices u and v there exists a path from u to v if and only if both coordinates of u are smaller than the corresponding coordinates of v.[3] The coordinates of such a drawing may also be used as a data structure that can be used to test whether one vertex of an st-planar graph can reach another in constant time per query.