Abstract
Recently a generalization of simple convex polytopes to combinatorial entities known as abstract polytopes has been proposed. The graph of an abstract polytope of dimensiond is a regular connected graph of degreed. Given a connected regular graph Г of degreed, it is interesting to find out whether it is the graph of some abstract polytopeP. We obtain necessary and sufficient conditions for this, in terms of the existence of a class of simple cycles in Г satisfying certain properties. The main result in this paper is that if a pair of simple convex polytopes or abstract polytopes have the same two-dimensional skeleton, then they are isomorphic. Every two-dimensional face of a simple convex polytope or an abstract polytope is a simple cycle in its graph. Given the graph of a simple convex polytope or an abstract polytope and the simple cycles in this graph corresponding to all its two-dimensional faces, then we show how to construct all its remaining faces. Given a regular connected graph Г and a class of simple cylesD in it, we provide necessary and sufficient conditions under whichD is the class of two-dimensional faces of some abstract polytope which has Г as its graph.
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References
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This research has been partially supported by the ISDOS Research Project at the Department of Industrial and Operations Engineering, and by the National Science Foundation under Grant No. GK-27872 with the University of Michigan.
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Murty, K.G. The graph of an abstract polytope. Mathematical Programming 4, 336–346 (1973). https://doi.org/10.1007/BF01584675
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DOI: https://doi.org/10.1007/BF01584675