Lattices and maximum flow algorithms in planar graphs

TL;DR

This paper explores the lattice structure of s-t-paths in planar graphs, showing how certain properties relate to maximum flow algorithms and characterizing s-t-planar graphs.

Contribution

It demonstrates that the s-t-paths form a submodular lattice in plane graphs and connects this to maximum flow algorithms, providing a new characterization of s-t-planar graphs.

Findings

s-t-paths form a submodular lattice in plane graphs

Ford-Fulkerson's algorithm is a special case of a greedy lattice algorithm in s-t-planar graphs

Submodularity and consecutivity cannot both hold in non-s-t-planar planar graphs

Abstract

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.