Obstructions to weak decomposability for simplicial polytopes
Open Access Publications from the University of California

## Obstructions to weak decomposability for simplicial polytopes

• Author(s): Hähnle, Nicolai
• Klee, Steven
• Pilaud, Vincent
• et al.

## Published Web Location

https://arxiv.org/pdf/1206.6143.pdf
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Abstract

Provan and Billera introduced notions of (weak) decomposability of simplicial complexes as a means of attempting to prove polynomial upper bounds on the diameter of the facet-ridge graph of a simplicial polytope. Recently, De Loera and Klee provided the first examples of simplicial polytopes that are not weakly vertex-decomposable. These polytopes are polar to certain simple transportation polytopes. In this paper, we refine their analysis to prove that these $d$-dimensional polytopes are not even weakly $O(\sqrt{d})$-decomposable. As a consequence, (weak) decomposability cannot be used to prove a polynomial version of the Hirsch conjecture.

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