Graph recognition of 4-dimensional polytopes from 1-skeletons

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

Background

Steinitz’s theorem characterizes 3D polytope graphs (planar, 3-connected). In 4D, realization and graph recognition become far more complex, and universality suggests hardness.

The authors indicate that the 4D graph recognition problem remains open.

References

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 Compendium — Problem '4-dimensional Polytope Realizability'