Examples and Counterexamples for the Perles Conjecture
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The combinatorial structure of a d-dimensional simple convex polytope—as given, for example, by the set of the (d − 1)-regular subgraphs of the facets—can be reconstructed from its abstract graph. However, no polynomial/efficient algorithm is known for this task, although a polynomially checkable certificate for the correct reconstruction exists.
A much stronger certificate would be given by the following characterization of the facet subgraphs, conjectured by Micha Perles: “The facet subgraphs of a simple d-polytope are exactly all the (d − 1)-regular, connected, induced, non-separating subgraphs.”
We present non-trivial classes of examples for the validity of the Perles conjecture: in particular, it holds for the duals of cyclic polytopes, and for the duals of stacked polytopes.
On the other hand, we observe that for any 4-dimensional counterexample, the boundary of the (simplicial) dual polytope P Δ contains a 2-complex without a free edge, and without 2-dimensional homology. Examples of such complexes are known; we use a modification of “Bing’s house” (two walls removed) to construct explicit 4-dimensional counterexamples to the Perles conjecture.