DOI: 10.1016/0097-3165(88)90064-7 URL: https://linkinghub.elsevier.com/retrieve/pii/0097316588900647 Authors: Gil Kalai Journal/publisher: Year: 1988
Title: A simple way to tell a simple polytope from its graph
Abstract: Let P be a simple d-dimensional polytope and let G(P) be the graph of P. Thus, G(P) is an abstract graph defined in the set of vertices V(P) of P. Two vertices u and u in V(P) are adjacent in G(P) if [u, u] is a l-dimensional face of P. Perles [P] conjectured and Blind and Mani [BM] recently proved that G(P) determines the entire combinatorial structure of P. Here is a simple proof of this result. Let f denote the number of non-empty faces of P. We consider the class of acylic orientations (i.e., edge orientations with no oriented cycles) of G(P). We will not distinguish between an acyclic orientation 0 of G(P) and the partial order induced by 0 on V(P).