The graph of an abstract polytope

Recently a generalization of simple convex polytopes to combinatorial entit ies known as abstract polytopes has been proposed. The graph of an abstract polytope o f d imension d is a regular connected graph of degree d. Given a connected regular graph F o f degree d, it is interesting to find out whether it is the graph of some abstract polytope P. We obtain necessary and sufficient condit ions for this, in terms of the existence of a class o f simple cycles in P satisfying certain properties. The main result in this paper is that if a pair o f simple convex polytopes or abstract polytopes have the same two-dimensional skeleton, then they are isomorphic. Every twodimensional 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 F and a class o f simple cyles D ,in it, we provide necessary and sufficient condit ions under which ~ i s the class of two-dimensional faces of some abstract polytope which has r as its graph.