A simple way to tell a simple polytope from its graph
Journal of Combinatorial Theory Series A
Computing the face lattice of a polytope from its vertex-facet incidences
Computational Geometry: Theory and Applications
Unique Sink Orientations of Cubes
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Reconstructing a simple polytope from its graph
Combinatorial optimization - Eureka, you shrink!
Unique sink orientations of grids
IPCO'05 Proceedings of the 11th international conference on Integer Programming and Combinatorial Optimization
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We show that one can compute a (simple) polytope from its graph in Polynomial time. This computation of a polytope from its graph was shown to be solvable by Blind and Mani and more recently Kalai provided a simple proof that leads to an exponential time algorithm. Our proof relies on a Primal-Dual characterization by Joswig, Kaibel and Korner. We describe an exponential Linear Programming which can be used to construct the solution and show that it can be solved in polynomial time.