Consider the following line of reasoning that shows certain simplicial complexes (of arbitrary dimension) are completely determined by corresponding graphs:


1. The facet complex of any [simplicial polytope](https://en.wikipedia.org/wiki/Simplicial_polytope) is a simplicial complex.
2. The facet complex completely determines (and is completely determined by) the original simplicial polytope.
3. The simplicial polytope is dual to a [***simple*** polytope](https://en.wikipedia.org/wiki/Simple_polytope).
4. The dual ***simple*** polytope completely determines (and is completely determined by) the original simplicial polytope.
5. By a Theorem of Whitney, cf. the references [a] and [b] below, the dual ***simple*** polytope is completely determined by (and completely determines) its 1-skeleton.



> 
> **Question:** For which simplicial complexes does the above chain of reasoning generalize?
> 
> 
> In other words, what is a necessary and sufficient characterization of simplicial complexes with the property that their (polyhedrally) dual polyhedral complexes are completely determined by the 1-skeleton of the dual?
> 
> 
> Equivalently, which polyhedral complexes generalize ***simple*** polytopes in having the two properties that (1) they are (polyhedrally) dual to a simplicial complex, and (2) they are completely determined by their 1-skeleton?
> 
> 
> 


**Note:** Feel free to assume all involved simplicial complexes, cell complexes, and graphs have finitely many vertices.


**Note:** This is a [cross-post of an unanswered question](https://math.stackexchange.com/questions/4715194/which-simplicial-complexes-are-completely-determined-by-the-1-skeleton-of-their) on math.stackexchange. I'm not sure whether this is a research-level question, but it might be because it's asking for references about generalizations of results from the late '80's, in a field (combinatorial topology) that seems to be still actively researched. I am an outsider to this field myself, however, so I'm uncertain how basic this question is. (Likewise I wasn't sure which tags to add.)


**References**


**[a]** Blind, Roswitha; Mani-Levitska, Peter (1987), *"[Puzzles and polytope isomorphisms](https://link.springer.com/article/10.1007/BF01830678)"*, Aequationes Mathematicae, 34 (2–3): 287–297, doi:10.1007/BF01830678, MR 0921106


**[b]** Kalai, Gil (1988), *"[A simple way to tell a simple polytope from its graph](https://www.sciencedirect.com/science/article/pii/0097316588900647?via%3Dihub)"*, Journal of Combinatorial Theory, Series A, 49 (2): 381–383, doi:10.1016/0097-3165(88)90064-7, MR 0964396


**Related, but distinct, questions on MathOverflow:** Unlike this related question [Cyclic polytopes whose boundary is a flag complex](https://mathoverflow.net/questions/266009/cyclic-polytopes-whose-boundary-is-a-flag-complex) I am not asking about whether the simplicial complex is determined by the simplicial complex's 1-skeleton, I am asking about whether its dual polyhedral complex is determined by its dual polyhedral complex's 1-skeleton.


This other question ([Visualizing polyhedra from their 1-skeletons](https://mathoverflow.net/questions/119455/visualizing-polyhedra-from-their-1-skeletons)) also appears to be (indirectly related), but I am asking specifically about the level of generality of arbitrary polyhedral complexes, not only inidividual polyhedra (for which the answer is already known and given in the two references below).


This question ([Can I build infinitely many polytopes from only finitely many prescribed facets?](https://mathoverflow.net/questions/351241/can-i-build-infinitely-many-polytopes-from-only-finitely-many-prescribed-facets)) is asking about a case where the maximal polyhedral cells of a (single polytope considered as a pure) polyhedral complex are restricted to belonging to a certain family of polyhedra, but does not seem to require that the *facet complex* itself be specified a priori (the latter of which is what I am asking). So it also seems to be a different question.


(Also technically I'm not certain that the facet complex would uniquely specify the original polytope, the converse is trivial of course, but I also don't see how it wouldn't.)



Which simplicial complexes are completely determined by the 1-skeleton of their dual polyhedral complexes?
A simplicial polytope is determined by the 1-skeleton of its dual *as
a polytope*, not as a simplicial complex. For instance, take a
sufficiently large simplicial polytope of dimension at least three,
and identify two far away vertices (so that it remains a simplicial
complex). The dual polyhedral complexes of these two simplicial
complexes have the same 1-skeleton.


As suggested by Sam Hopkins, we can ask for families of simplicial
complexes whose members *within the family* are determined by their
1-skeleton. Possible families, in increasing order of generality, are
triangulations of spheres, Gorenstein\* complexes, and doubly
Cohen-Macaulay complexes. On the other hand, the family of
Cohen-Macaulay complexes is too general. There are two nonisomorphic
triangulations of a 2-dimensional ball with four facets (triangles)
whose 1-skeletons are both paths of length three.


