Abstract
We investigate the vertex-connectivity of the graph of f -mono- tone 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.
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