Gruber P. Convex and Discrete Geometry

276 Convex Polytopes
The existence of good acyclic orientations of G(P) is easy to see. Choose u ∈
Sd−1 such that the linear form x → u·x assumes different values at different vertices
of P. Then orient an edge vw of P from v to w if u · vu · w.
We now characterize, among all acyclic orientations of G(P), the good ones. Let
O be an acyclic orientation. The in-degree of a vertex v of G(P) is the number of
edges incident with v and oriented towards it. Let hO i be the number of vertices of
G(P) with in-degree i, where i = 0, 1,...,d, and let
f O = h O 0 + 2hO 1 + 22 h O 2 +···+2d h O d .
If a vertex v of G(P) has in-degree i, then v is a sink of 2 i faces of P. (Since P is
simple, any set of j ≤ i edges of P incident with v determines a face of dimension
j of P which contains these edges but no further edge incident with v.) Let f be the
number of non-empty faces of P. Since each face of P has at least one sink,
Thus,
f O ≥ f, and O is good if and only if f O = f.
(3) among all acyclic orientations O of G(P), the good orientations are precisely
those with minimum f O .
It is easy to see that each face F of the simple polytope P is simple in aff F,
i.e. F is i-regular, where i = dim F. Clearly, G(F) is a sub-graph of G(P). Wenext
characterize among all sub-graphs of G(P) those which are the edge graphs of faces
of P:
(4) Let H be an induced sub-graph of G(P) that is, it contains all edges of G(P)
which are incident only with vertices of H. Then the following statements
are equivalent:
(i) H = G(F) where F is a face of P.
(ii) H is connected, i-regular for some i and initial with respect to some
good acyclic orientation O of G(P).
Here, when saying that the induced sub-graph H is initial, we mean that each edge
of G(P) which is incident with precisely one vertex of H (and thus is not an edge of
H) is oriented away from this vertex.
(i)⇒(ii) Clearly, H is connected and i-regular where i = dim F. Letu ∈ S d−1
such that the linear function x → u · x assumes different values at different vertices
of P and such that the values at the vertices of F are smaller than the values at the
other vertices of P. The orientation O which is determined by this linear form is an
acyclic good orientation and H is initial with respect to it.
(ii)⇒(i) Let v beasinkofH with respect to O. There are i edges in H incident
with v and oriented towards it. Thus v is a sink of the face F of P of dimension
i containing these i edges. Since O is good, v is the unique sink of F and thus all
vertices of F are �O v. Being initial, H contains all vertices �O v. Hence V(F) ⊆