Examples and counterexamples for Perles' conjecture

TL;DR

This paper investigates Perles' conjecture on the structure of facet subgraphs in simple convex polytopes, providing examples where it holds and constructing explicit counterexamples in four dimensions.

Contribution

It confirms Perles' conjecture for duals of cyclic and stacked polytopes and constructs explicit 4D counterexamples using topological obstructions.

Findings

Perles' conjecture holds for duals of cyclic polytopes.

Perles' conjecture holds for duals of stacked polytopes.

Explicit 4D counterexamples are constructed using topological obstructions.

Abstract

The combinatorial structure of a d-dimensional simple convex polytope can be reconstructed from its abstract graph [Blind & Mani 1987, Kalai 1988]. However, no polynomial/efficient algorithm is known for this task, although a polynomially checkable certificate for the correct reconstruction was found by [Joswig, Kaibel & Koerner 2000]. A much stronger certificate would be given by the following characterization of the facet subgraphs, conjectured by M. Perles: ``The facet subgraphs of the graph of a simple d-polytope are exactly all the (d-1)-regular, connected, induced, non-separating subgraphs'' [Perles 1970]. We give examples for the validity of Perles conjecture: In particular, it holds for the duals of cyclic polytopes, and for the duals of stacked polytopes. On the other hand, we identify a topological obstruction that must be present in any counterexample to Perles' conjecture; thus, starting with a modification of ``Bing's house'', we construct explicit 4-dimensional counterexamples.