Graph-Only Recognition of 4D Polytopes

Determine the computational complexity of recognizing whether a given graph is the 1-skeleton of a 4-dimensional polytope.

For 3D polytopes, Steinitz’s theorem gives a simple graph characterization, but no such tractable characterization is known for 4D.

The survey notes that recognizing 4D polytopes from their graphs (1-skeletons) remains unresolved.

The complexity of the related problem in dimension 4 in which we are only given the graph ($1$-skeleton) of the polytope is also open.

— The Existential Theory of the Reals as a Complexity Class: A Compendium (2407.18006 - Schaefer et al., 25 Jul 2024) in ProblemCG “4-dimensional Polytope Realizability” entry