Reconstructing a Simple Polytope from its Graph
摘要
Blind and Mani (1987) proved that the entire combinatorial structure (the vertex-facet incidences) of a simple convex polytope is determined by its abstract graph. Their proof is not constructive. Kalai (1988) found a short, elegant, and algorithmic proof of that result. However, his algorithm has always exponential running time. We show that the problem to reconstruct the vertex-facet incidences of a simple polytope P from its graph can be formulated as a combinatorial optimization problem that is strongly dual to the problem of finding an abstract objective function on P (i.e., a shelling order of the facets of the dual polytope of P). Thereby, we derive polynomial certificates for both the vertex-facet incidences as well as for the abstract objective functions in terms of the graph of P. The paper is a variation on joint work with Michael Joswig and Friederike Koerner (2001).
引用
@article{arxiv.math/0202103,
title = {Reconstructing a Simple Polytope from its Graph},
author = {Volker Kaibel},
journal= {arXiv preprint arXiv:math/0202103},
year = {2007}
}
备注
14 pages