Abstract

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

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