Discrete & Computational Geometry
Theory of linear and integer programming
Theory of linear and integer programming
American Mathematical Monthly
The lattice structure of flow in planar graphs
SIAM Journal on Discrete Mathematics
The lattice structure of the set of domino tilings of a polygon
Theoretical Computer Science - Discrete applied problems, florilegium for E. Goles
On the Complexity of Computing the Tutte Polynomial of Bicircular Matroids
Combinatorics, Probability and Computing
Discrete & Computational Geometry
Combinatorics, Probability and Computing
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A D-polyhedron is a polyhedron P such that if x,y are in P then so are their componentwise maximums and minimums. In other words, the point set of a D-polyhedron forms a distributive lattice with the dominance order. We provide a full characterization of the bounding hyperplanes of D-polyhedra. Aside from being a nice combination of geometric and order theoretic concepts, D-polyhedra are a unifying generalization of several distributive lattices which arise from graphs. In fact with a D-polyhedron we associate a directed graph with arc-parameters, such that points in the polyhedron correspond to vertex potentials on the graph. Alternatively, an edge-based description of the points of a D-polyhedron can be given. In this model the points correspond to the duals of generalized flows, i.e., duals of flows with gains and losses. These models can be specialized to yield distributive lattices that have been previously studied. Particular specializations are: flows of planar digraphs (Khuller, Naor and Klein), @a-orientations of planar graphs (Felsner), c-orientations (Propp) and @D-bonds of digraphs (Felsner and Knauer). As an additional application we identify a distributive lattice structure on generalized flow of breakeven planar digraphs.