Abstract
We show that the left/right relation on the set of s-t-paths of a plane graph [9] 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 [4] is indeed a special case of a two-phase greedy algorithm on lattice polyhedra [2]. 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.
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Matuschke, J., Peis, B. (2010). Lattices and Maximum Flow Algorithms in Planar Graphs. In: Thilikos, D.M. (eds) Graph Theoretic Concepts in Computer Science. WG 2010. Lecture Notes in Computer Science, vol 6410. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16926-7_30
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DOI: https://doi.org/10.1007/978-3-642-16926-7_30
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