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22 votes
4 answers
3k views

Can you determine whether a graph is the 1-skeleton of a polytope?

How do I test whether a given undirected graph is the 1-skeleton of a polytope? How can I tell the dimension of a given 1-skeleton?
Hans-Peter Stricker's user avatar
16 votes
3 answers
720 views

Can we realize a graph as the skeleton of a polytope that has the same symmetries?

Given a graph $G$, a realization of $G$ as a polytope is a convex polytope $P\subseteq \Bbb R^n$ with $G$ as its 1-skeleton. A realization $P\subseteq \Bbb R^n$ is said to realize the symmetries of $...
M. Winter's user avatar
  • 13.6k
18 votes
2 answers
573 views

Can the graph of a symmetric polytope have more symmetries than the polytope itself?

I consider convex polytopes $P\subseteq\Bbb R^d$ (convex hull of finitely many points) which are arc-transitive, i.e. where the automorphism group acts transitively on the 1-flags (incident vertex-...
M. Winter's user avatar
  • 13.6k
10 votes
3 answers
500 views

Given the skeleton of an inscribed polytope. If I move the vertices so that no edge increases in length, can the circumradius still get larger?

Let $P\subset \Bbb R^n$ be an inscribed convex polytope, that is, all its vertices are on a common sphere of radius $r$. Let $G$ be the edge-graph of $P$. For convenience, assume $V(G)=\{1,\dotsc,s\}$....
M. Winter's user avatar
  • 13.6k
8 votes
2 answers
392 views

Can two non-equivalent polytopes of same dimension have the same graph?

By a polytope I mean the convex hull of finitely many points. The graph of a polytope is the graph isomorphic to its 1-skeleton. By equivalence of polytopes I mean combinatorial equivalence, i.e. ...
M. Winter's user avatar
  • 13.6k
4 votes
0 answers
132 views

Can a polytopal graph be "centrally symmetric" in more than one way?

Let $P,Q$ be two centrally symmetric convex polytopes, potentially of different dimensions and combinatorial type, but with the same edge-graph $G$. The central symmetry of $P$ induces an involutory ...
M. Winter's user avatar
  • 13.6k
2 votes
0 answers
126 views

Are there half-transitive convex polytopes?

I only consider convex polytopes, i.e. convex hulls of finitely many points. The (edge-)graph of a polytope $P\subseteq\Bbb R^d$ is the graph consisting of the polytope's vertices, two are adjacent if ...
M. Winter's user avatar
  • 13.6k