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Enumerative problems for arborescences and monotone paths on polytope graphs

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.