Monotone paths on polytopes
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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 also re-examine the Baues problem for cellular strings on polytopes, solved by Billera, Kapranov and Sturmfels. Our analysis shows that their positive result is a direct consequence of shellability of polytopes and is therefore less related to convexity than is at first apparent.
- Monotone paths on polytopes
Volume 235, Issue 2 , pp 315-334
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- A1. Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden (e-mail: email@example.com), SE
- A2. School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA (e-mail: firstname.lastname@example.org; email@example.com), US