In a unique-sink orientation of a simple polytope, every subset of k incoming edges at a vertex v determines a k-dimensional face for which v is the unique sink.
Therefore, the number of faces of all dimensions of the polytope (including the polytope itself, but not the empty set) can be computed by the sum of the number of subsets of incoming edges, where G(P) is the graph of the polytope, and din(v) is the in-degree (number of incoming edges) of a vertex v in the given orientation (Kalai 1988).
Additionally, a k-regular subgraph of the given graph forms a face of the polytope if and only if its vertices form a lower set for at least one acyclic unique sink orientation.
In this way, the face lattice of the polytope is uniquely determined from the graph (Kalai 1988).
Based on this structure, the face lattices of simple polytopes can be reconstructed from their graphs in polynomial time using linear programming (Friedman 2009).