reconstructing a simple polytope from its graph

Question

I'm reading Kalai's paper on the reconstruction of simple polytopes from their graphs and there is something I don't understand.

we define a good acyclic orientation as an acyclic orientation for which every non empty face $F$ of $P$, $G(F)$ has exactly one sink. Take $O$ such an orientation.

define $h_k^O$ as the number of vertices of index $k$ in $G(P)$ and define $$f^O = h_0^O + 2h_1^O + \dots + 2^dh^O_d$$

Then it states that $f^O\geqslant f$ ($f$ the number of faces) and $O$ is good iff $f^0 = f$. I don't get these two implications. It should be easy but I don't see it.

PS: a link to the paper: http://www.sciencedirect.com/science/article/pii/0097316588900647

EDIT: Problem solved

Answer 1

I found the answer on the following link

https://gilkalai.wordpress.com/2009/01/16/telling-a-simple-polytope-from-its-graph/