Monotone paths on polytopes

We investigate the vertex-connectivity of the graph of f-monotone paths on a cl-polytope P with respect to a generic functional f. The third author ha s conjectured that this graph is always (d - 1)-connected. We resolve this conjecture positively for simple polytopes and show that the graph is 2-con nected for any d-polytope with d greater than or equal to 3. However,we dis prove the conjecture in general by exhibiting counterexamples for each d gr eater than or equal to 4 in which the graph has a vertex of degree two. We also re-examine the Baues problem for cellular strings on polytopes, sol ved by Billera, Kapranov and Sturmfels. Our analysis shows that their posit ive result is a direct consequence of shellability of polytopes and is ther efore less related to convexity than is at first apparent.