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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 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.
  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 dual cell complexes (a.k.a. "polyhedral complexes") are completely determined by the 1-skeleton of the dual?

Equivalently, which cell complexes generalize simple polytopes in having the two properties that (1) they are dual to a simplicial complex, and (2) they are completely determined by their 1-skeleton?

Note: Here I am using terminology where "(combinatorial) cell complex" is synonymous with "(abstract / combinatorial) polyhedral complex", because I have seen that usage in several sources.

("Cell complex" can have quite a different meaning in other sources, cf. here or here.)

Note: The "duality" that this question refers to specifically is "polyhedral duality", which some sources seem to call either "topological duality" or "Poincare duality".

Given a $d$-dimensional "polyhedral complex"/"cell complex", every $k$-cell is swapped with a $(d-k)$-cell.

My understanding is that a "polyhedral complex" can also be considered as a poset (in a way generalizing how a polytope can be considered a lattice via the face lattice), and that from this perspective "polyhedral duality" is just "poset duality" (which in turn can be considered a special case of categorical duality, but anyway).

E.g. simplicial complexes can also be considered special cases of hypergraphs, but the duality this question asks about is not hypergraph duality. Likewise, I am also not asking about Alexander/Alexandrov duality of simplicial complexes.

Note: Feel free to assume all involved simplicial complexes, cell complexes, and graphs have finitely many vertices. (I think that is sufficient to ensure the simplicial complexes involved are compact and orientable and thus have well-defined dual polyhedral complexes?)

References

[a] Blind, Roswitha; Mani-Levitska, Peter (1987), "Puzzles and polytope isomorphisms", 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", Journal of Combinatorial Theory, Series A, 49 (2): 381–383, doi:10.1016/0097-3165(88)90064-7, MR 0964396

Possibly related questions:
Average degree of graph from polyhedral complex
Poincare dual simplicial structure of complexes homotopy equivalent to manifolds
Does there exist a class of graphs with $O(n^2)$ maximal cliques which can be realized as the 1-skeleton of a Vietoris-Rips complex in $\mathbb R^2$?

hasManyStupidQuestions
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  • To quote from our guidelines for formatting and writing your question, Don't force someone to click on an external link just to see or understand your question. In particular, trying to understand the definitions of your main terminology by following those wikipedia links is something of a wild goose chase. Please define your terms in your post. – Lee Mosher Jun 08 '23 at 22:49
  • @LeeMosher Which terms should be defined? This area is not my specialty, so I assumed the definitions of all terms would be common knowledge for anyone able to answer the question. (E.g. most questions about calculus don't provide definitions of "real number", "derivative", "function", etc. The line has to be drawn somewhere, but I'm not sure where the line should be drawn here.) – hasManyStupidQuestions Jun 08 '23 at 22:57

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