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.
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