The graph of an abstract polytope

Recently a generalization of simple convex polytopes to combinatorial entities known as ab-stract polytopes has been proposed. The graph of an abstract polytope of dimension d is a regu-lar connected graph of degree d. Given a connected regular graph F of degree d, it is interesting to find out whether it is the graph of some abstract polytope P. We obtain necessary and suffi-cient conditions for this, in terms of the existence of a class of simple cycles in P satisfying cer-tain properties. The main result in this paper is that if a pair of simple convex polytopes or ab-stract 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 con-struct all its remaining faces. Given a regular connected graph F and a class of simple cyles D,in it, we provide necessary and sufficient conditions under which ~is the class of two-dimensional faces of some abstract polytope which has r as its graph. 1