Monotone paths on polytopes

@article{Athanasiadis2000MonotonePO,
  title={Monotone paths on polytopes},
  author={Christos A. Athanasiadis and Paul H. Edelman and Victor Reiner},
  journal={Mathematische Zeitschrift},
  year={2000},
  volume={235},
  pages={315-334}
}
Abstract. We investigate the vertex-connectivity of the graph of f-monotone paths on a d-polytopeP with respect to a generic functionalf. The third author has conjectured that this graph is always (d $-1$)-connected. We resolve this conjecture positively for simple polytopes and show that the graph is 2-connected for any d-polytope with $d \geq 3$. However, we disprove the conjecture in general by exhibiting counterexamples for each $d \geq 4$ in which the graph has a vertex of degree two.We… 

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