  We show that the left/right relation on the set of s-t-paths of a plane graph
induces a so-called submodular lattice. If the embedding of the graph is
s-t-planar, this lattice is even consecutive. This implies that Ford and
Fulkerson's uppermost path algorithm for maximum flow in such graphs is indeed
a special case of a two-phase greedy algorithm on lattice polyhedra. We also
show that the properties submodularity and consecutivity cannot be achieved
simultaneously by any partial order on the paths if the graph is planar but not
s-t-planar, thus providing a characterization of this class of graphs.
