On thek-systems of a simple polytope

Ak-system of the graphG _P of a simple polytopeP is a set of induced subgraphs ofG _P that shares certain properties with the set of subgraphs induced by thek-faces ofP. This new concept leads to polynomial-size certificates in terms ofG _P for both the set of vertex sets of facets and for abstract objective functions (AOF) in the sense of Kalai. Moreover, it is proved that an acyclic orientation yields an AOF if and only if it induces a unique sink on every 2-face.