Monotone paths on polytopes

We investigate the vertex-connectivity of the graph of f-monotone paths on a d-polytope P with respect to a generic functional f. 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 ≥ 3. However, we disprove the conjecture in general by exhibiting counterexamples for each d ≥ 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.