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Every generic linear functional $f$ on a convex polytope $P$ induces anorientation on the graph of $P$. From the resulting directed graph one candefine a notion of $f$-arborescence and $f$-monotone path on $P$, as well as anatural graph structure on the vertex set of $f$-monotone paths. These conceptsare important in geometric combinatorics and optimization.  This paper bounds the number of $f$-arborescences, the number of $f$-monotonepaths, and the diameter of the graph of $f$-monotone paths for polytopes $P$ interms of their dimension and number of vertices or facets.

Enumerative problems for arborescences and monotone paths on polytope graphs