  Every generic linear functional $f$ on a convex polytope $P$ induces an
orientation on the graph of $P$. From the resulting directed graph one can
define a notion of $f$-arborescence and $f$-monotone path on $P$, as well as a
natural graph structure on the vertex set of $f$-monotone paths. These concepts
are important in geometric combinatorics and optimization.
  This paper bounds the number of $f$-arborescences, the number of $f$-monotone
paths, and the diameter of the graph of $f$-monotone paths for polytopes $P$ in
terms of their dimension and number of vertices or facets.
