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### Reconstructing a Simple Polytope from its Graph [article]

Volker Kaibel
2002 arXiv   pre-print
We show that the problem to reconstruct the vertex-facet incidences of a simple polytope P from its graph can be formulated as a combinatorial optimization problem that is strongly dual to the problem  ...  Blind and Mani (1987) proved that the entire combinatorial structure (the vertex-facet incidences) of a simple convex polytope is determined by its abstract graph. Their proof is not constructive.  ...  Every face of a simple polytope is simple as well. None of the examples showing that it is, in general, impossible to reconstruct the face lattice of a polytope from its graph, is simple.  ...

### Reconstructing a non-simple polytope from its graph [article]

Michael Joswig
1999 arXiv   pre-print
A well-known theorem of Blind and Mani says that every simple polytope is uniquely determined by its graph.  ...  We apply our results to a special class of cubical polytopes.  ...  The polytope pictures have been produced with Geomview  and Graphlet  via polymake  .  ...

### Polytopes close to being simple [article]

Guillermo Pineda-Villavicencio, Julien Ugon, David Yost
2018 arXiv   pre-print
It is known that polytopes with at most two nonsimple vertices are reconstructible from their graphs, and that d-polytopes with at most d-2 nonsimple vertices are reconstructible from their 2-skeletons  ...  Simple polytopes are those with excess zero. We prove that polytopes with excess at most d-1 are reconstructible from their graphs, and this is best possible.  ...  A polytope can be reconstructed from its graph if the valid frames of each vertex are known.  ...

### Reconstructing Nearly Simple Polytopes from their Graph [article]

Joseph Doolittle
2017 arXiv   pre-print
We also give an example of a 3-nearly simple polytope which is not reconstructible from its graph.  ...  We present a partial description of which polytopes are reconstructible from their graphs.  ...  A 0,1, or 2-nearly simple convex polytope can be reconstructed from its graph. There exist 3-nearly simple convex polytopes which can not be reconstructed from their graphs.  ...

### Graphs, Skeleta and Reconstruction of Polytopes [article]

Margaret M. Bayer
2017 arXiv   pre-print
A renowned theorem of Blind and Mani, with a constructive proof by Kalai and an efficiency proof by Friedman, shows that the whole face lattice of a simple polytope can be determined from its graph.  ...  This is part of a broader story of reconstructing face lattices from partial information, first considered comprehensively in Gr\"unbaum's 1967 book.  ...  Acknowledgments Thanks to Karim Adiprasito, Joseph Doolittle, Eran Nevo, Isabella Novik, Raman Sanyal, Hailun Zheng and Günter Ziegler for close reading of a draft and useful comments.  ...

### Reconstructibility of matroid polytopes [article]

Guillermo Pineda-Villavicencio, Benjamin Schröter
2021 arXiv   pre-print
We specify what is meant for a polytope to be reconstructible from its graph or dual graph.  ...  Additionally, we prove that matroid polytopes are class reconstructible from their graphs, and we present a O(n^3) algorithm that computes the vertices of a matroid polytope from its n-vertex graph.  ...  Matroids from hypersimplex splits. J. Combin. Theory Ser. A, 151:254–284, 2017. [Kal88] Gil Kalai. A simple way to tell a simple polytope from its graph. J. Combin. Theory Ser.  ...

### Examples and Counterexamples for the Perles Conjecture

Haase, Ziegler
2002 Discrete & Computational Geometry
The combinatorial structure of a d-dimensional simple convex polytope -as given, for example, by the set of the (d − 1)-regular subgraphs of facets -can be reconstructed from its abstract graph  [10  ...  Perles: "The facet subgraphs of a simple d-polytope are exactly all the (d − 1)-regular, connected, induced, non-separating subgraphs"  .  ...  Introduction If P is a d-dimensional simple polytope, then its graph G = G(P ) is a d-regular, d-connected graph.  ...

### On the reconstruction of polytopes [article]

Joseph Doolittle, Eran Nevo, Guillermo Pineda-Villavicencio, Julien Ugon, David Yost
2018 arXiv   pre-print
Blind and Mani, and later Kalai, showed that the face lattice of a simple polytope is determined by its graph, namely its 1-skeleton.  ...  Call a vertex of a d-polytope nonsimple if the number of edges incident to it is more than d.  ...  Reconstruction from graphs In this section we prove that, like simple polytopes, d-polytopes with at most two nonsimple vertices are reconstructible from their graphs; see Theorem 4.8.  ...

### On the Reconstruction of Polytopes

Joseph Doolittle, Eran Nevo, Guillermo Pineda-Villavicencio, Julien Ugon, David Yost
2018 Discrete & Computational Geometry
Blind and Mani, and later Kalai, showed that the face lattice of a simple polytope is determined by its graph, namely its 1-skeleton.  ...  Call a vertex of a d-polytope nonsimple if the number of edges incident to it is more than d.  ...  Reconstruction from graphs In this section we prove that, like simple polytopes, d-polytopes with at most two nonsimple vertices are reconstructible from their graphs; see Theorem 4.8.  ...

### Examples and counterexamples for Perles' conjecture [article]

Christian Haase, Günter M. Ziegler
2001 arXiv   pre-print
The combinatorial structure of a d-dimensional simple convex polytope can be reconstructed from its abstract graph [Blind & Mani 1987, Kalai 1988].  ...  Perles: "The facet subgraphs of the graph of a simple d-polytope are exactly all the (d-1)-regular, connected, induced, non-separating subgraphs" [Perles 1970].  ...  Introduction If P is a d-dimensional simple polytope, then its graph G = G(P ) is a d-regular, d-connected graph.  ...

### Page 4043 of Mathematical Reviews Vol. , Issue 2001F [page]

2001 Mathematical Reviews
a simple polytope from its graph.  ...  {For the entire collection see MR 2001d:52002.} 20011:52026 2001f:52023 52B11 Joswig, Michael (D-TUB; Berlin) Reconstructing a non-simple polytope from its graph.  ...

### A minimal counterexample to a strengthening of Perles' conjecture [article]

Joseph Doolittle
2018 arXiv   pre-print
The example is a simple 4-polytope that has an induced 3-connected 3-regular subgraph, whose graph complement is connected. This subgraph is planar and not the graph of a facet of the polytope.  ...  In this paper, we present a minimal counterexample to a conjecture of Perles that answers a question of Haase and Ziegler.  ...  If the conjecture were true, then simple polytopes could be reconstructed from their vertex-edge graphs.  ...

### Page 1005 of Mathematical Reviews Vol. 40, Issue 5 [page]

1970 Mathematical Reviews
This paper is primarily a review of known results on reconstruction of graphs. It includes many proofs, some of them rather different from those in the literature.  ...  channel from its topology.  ...

### On the k-Systems of a Simple Polytope [article]

Michael Joswig, Volker Kaibel, Friederike K"orner
2001 arXiv   pre-print
A k-system of the graph G(P) of a simple polytope P is a set of induced subgraphs of G(P) that shares certain properties with the set of subgraphs induced by the k-faces of P.  ...  Moreover, it is proved that an acyclic orientation yields an AOF if and only if it induces a unique sink on every 2-face.  ...  Thus, the complexity status of the problem of reconstructing the combinatorial type of a simple polytope from its graph remains unclear.  ...

### Eigenpolytopes, Spectral Polytopes and Edge-Transitivity [article]

Martin Winter
2020 arXiv   pre-print
Starting from a finite simple graph G, for each eigenvalue θ of its adjacency matrix one can construct a convex polytope P_G(θ), the so called θ-eigenpolytop of G.  ...  For some polytopes this technique can be used to reconstruct the polytopes from its edge-graph. Such polytopes (we shall call them spectral) are still badly understood.  ...  The edge-graph of a general polytope carries little information about that polytope i.e., given only its edge-graph, we can often not reconstruct the polytope from this (up to combinatorial equivalence  ...
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