![Page 1: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/1.jpg)
“Steinitz Theorems for Orthogonal Polyhedra” byDavid Eppstein and Elena Mumford (SoCG 2010)
Vinayak Pathak
July 25, 2010
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 2: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/2.jpg)
Skeleton graphs of polytopes
I Imagine a polytope.
I “Polytope is the general term of the sequence, point,segment, polygon, polyhedron, ... .” -H.S.M. Coxeter, inRegular Polytopes
I Skeleton graph is a graph whose nodes are the nodes of thepolytope and there is an edge between two nodes if thereexists a 1-face that connects them.
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 3: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/3.jpg)
Skeleton graphs of polytopes
I Imagine a polytope.
I “Polytope is the general term of the sequence, point,segment, polygon, polyhedron, ... .” -H.S.M. Coxeter, inRegular Polytopes
I Skeleton graph is a graph whose nodes are the nodes of thepolytope and there is an edge between two nodes if thereexists a 1-face that connects them.
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 4: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/4.jpg)
Skeleton graphs of polytopes
I Imagine a polytope.
I “Polytope is the general term of the sequence, point,segment, polygon, polyhedron, ... .” -H.S.M. Coxeter, inRegular Polytopes
I Skeleton graph is a graph whose nodes are the nodes of thepolytope and there is an edge between two nodes if thereexists a 1-face that connects them.
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 5: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/5.jpg)
Skeleton graphs of polytopes
Figure: The skeleton of an Icosidodecahedron
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
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Skeleton graphs of polytopes
Figure: The skeleton of a four dimensional cube
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
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Skeleton graphs of polytopes
Figure: The skeleton of a complicated four dimensional polytope
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 8: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/8.jpg)
Skeleton graphs of polytopes
I What do these graphs look like?
I What are their properties?
I Given a graph, can we decide if it is the skeleton graph ofsome polytope?
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 9: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/9.jpg)
Skeleton graphs of polytopes
I Good news: It is decidable using Tarski’s algorithm for realclosed fields.
I Bad news: No efficient algorithm is known.
I Slightly good news: The skeleton graph of a d dimensionalconvex polytope is always d-connected (Balinski’s Theorem).
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 10: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/10.jpg)
Skeleton graphs of polytopes
I Good news: It is decidable using Tarski’s algorithm for realclosed fields.
I Bad news: No efficient algorithm is known.
I Slightly good news: The skeleton graph of a d dimensionalconvex polytope is always d-connected (Balinski’s Theorem).
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 11: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/11.jpg)
Skeleton graphs of polytopes
I Good news: It is decidable using Tarski’s algorithm for realclosed fields.
I Bad news: No efficient algorithm is known.
I Slightly good news: The skeleton graph of a d dimensionalconvex polytope is always d-connected (Balinski’s Theorem).
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 12: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/12.jpg)
Polytopes in small dimensions
I A graph is a skeleton graph of a two dimensional polytope (or2-polytope) if and only if it’s a cycle.
I A graph is a skeleton graph of a polyhedron if and only if. . . no one knows.
I It’s not even known whether K12 is the skeleton graph ofsome polyhedron.
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 13: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/13.jpg)
Polytopes in small dimensions
I A graph is a skeleton graph of a two dimensional polytope (or2-polytope) if and only if it’s a cycle.
I A graph is a skeleton graph of a polyhedron if and only if. . . no one knows.
I It’s not even known whether K12 is the skeleton graph ofsome polyhedron.
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 14: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/14.jpg)
Polytopes in small dimensions
I A graph is a skeleton graph of a two dimensional polytope (or2-polytope) if and only if it’s a cycle.
I A graph is a skeleton graph of a polyhedron if and only if. . .
no one knows.
I It’s not even known whether K12 is the skeleton graph ofsome polyhedron.
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 15: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/15.jpg)
Polytopes in small dimensions
I A graph is a skeleton graph of a two dimensional polytope (or2-polytope) if and only if it’s a cycle.
I A graph is a skeleton graph of a polyhedron if and only if. . . no one knows.
I It’s not even known whether K12 is the skeleton graph ofsome polyhedron.
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 16: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/16.jpg)
Polytopes in small dimensions
I A graph is a skeleton graph of a two dimensional polytope (or2-polytope) if and only if it’s a cycle.
I A graph is a skeleton graph of a polyhedron if and only if. . . no one knows.
I It’s not even known whether K12 is the skeleton graph ofsome polyhedron.
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 17: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/17.jpg)
Convex polyhedra
I Steinitz Theorem!A graph is the skeleton graph of a convex polyhedron if andonly if it’s 3-connected and planar.
Figure: Left: a convex polyhedron, Right: its 3-connected, planarskeleton
I Branko Grunbaum says that Steinitz’s theorem is “the mostimportant and deepest known result on 3-polytopes.”
I Obvious next question: What about polyhedra that are notconvex?
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
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Convex polyhedra
I Steinitz Theorem!
A graph is the skeleton graph of a convex polyhedron if andonly if it’s 3-connected and planar.
Figure: Left: a convex polyhedron, Right: its 3-connected, planarskeleton
I Branko Grunbaum says that Steinitz’s theorem is “the mostimportant and deepest known result on 3-polytopes.”
I Obvious next question: What about polyhedra that are notconvex?
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 19: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/19.jpg)
Convex polyhedra
I Steinitz Theorem!A graph is the skeleton graph of a convex polyhedron if andonly if it’s 3-connected and planar.
Figure: Left: a convex polyhedron, Right: its 3-connected, planarskeleton
I Branko Grunbaum says that Steinitz’s theorem is “the mostimportant and deepest known result on 3-polytopes.”
I Obvious next question: What about polyhedra that are notconvex?
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 20: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/20.jpg)
Convex polyhedra
I Steinitz Theorem!A graph is the skeleton graph of a convex polyhedron if andonly if it’s 3-connected and planar.
Figure: Left: a convex polyhedron, Right: its 3-connected, planarskeleton
I Branko Grunbaum says that Steinitz’s theorem is “the mostimportant and deepest known result on 3-polytopes.”
I Obvious next question: What about polyhedra that are notconvex?
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 21: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/21.jpg)
Convex polyhedra
I Steinitz Theorem!A graph is the skeleton graph of a convex polyhedron if andonly if it’s 3-connected and planar.
Figure: Left: a convex polyhedron, Right: its 3-connected, planarskeleton
I Branko Grunbaum says that Steinitz’s theorem is “the mostimportant and deepest known result on 3-polytopes.”
I Obvious next question: What about polyhedra that are notconvex?
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 22: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/22.jpg)
Simple orthogonal polyhedra
I “Orthogonal” means each face is perpendicular to one of thecoordinate axes.
I “Simple” means three things.
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 23: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/23.jpg)
Simple orthogonal polyhedra
I “Orthogonal” means each face is perpendicular to one of thecoordinate axes.
I “Simple” means three things.
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 24: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/24.jpg)
Simple orthogonal polyhedra
I “Orthogonal” means each face is perpendicular to one of thecoordinate axes.
I “Simple” means three things.
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 25: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/25.jpg)
Simple orthogonal polyhedra
It should have the topology of a sphere.
Figure: Not allowed.
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
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Simple orthogonal polyhedra
Exactly three mutually perpendicular edges should meet at eachvertex.
Figure: Not allowed.
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 27: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/27.jpg)
Simple orthogonal polyhedra
Each face should be simply connected.
Figure: Not allowed.
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 28: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/28.jpg)
Simple orthogonal polyhedra
Figure: Some simple orthogonal polyhedra. (Simple means - 1. shouldhave the topology of a sphere, 2. exactly three mutually perpendicularedges should meet at each vertex, 3. faces should be simply connected.)
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 29: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/29.jpg)
Observations
Figure: Some simple orthogonal polyhedra. (Simple means - 1. shouldhave the topology of a sphere, 2. exactly three mutually perpendicularedges should meet at each vertex, 3. faces should be simply connected.)
The skeleton must be
I planar
I bipartite
I 3-regular
I 2-connected
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 30: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/30.jpg)
Observations
Figure: Some simple orthogonal polyhedra. (Simple means - 1. shouldhave the topology of a sphere, 2. exactly three mutually perpendicularedges should meet at each vertex, 3. faces should be simply connected.)
The skeleton must be
I planar
I bipartite
I 3-regular
I 2-connected
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 31: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/31.jpg)
Observations
Figure: Some simple orthogonal polyhedra. (Simple means - 1. shouldhave the topology of a sphere, 2. exactly three mutually perpendicularedges should meet at each vertex, 3. faces should be simply connected.)
The skeleton must be
I planar
I bipartite
I 3-regular
I 2-connected
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 32: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/32.jpg)
Observations
Figure: Some simple orthogonal polyhedra. (Simple means - 1. shouldhave the topology of a sphere, 2. exactly three mutually perpendicularedges should meet at each vertex, 3. faces should be simply connected.)
The skeleton must be
I planar
I bipartite
I 3-regular
I 2-connected
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 33: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/33.jpg)
Does the converse hold?
Figure: A planar, bipartite, 3-regular and 2-connected graph that is notthe skeleton of a simple orthogonal polyhedron.
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 34: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/34.jpg)
xyz polyhedra
TheoremA graph is the skeleton graph of an xyz polyhedron if and only if itis
I planar
I bipartite
I 3-regular
I 3-connected
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
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xyz polyhedra
I An xyz polyhedron is a simple orthogonal polyhedron forwhich any axis-parallel line contains at most 2 vertices.
Figure: An xyz polyhedron.
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 36: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/36.jpg)
xyz polyhedra
Unfortunately, there are simple orthogonal polyhedra that are notxyz.
Figure: Not xyz.
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 37: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/37.jpg)
Main result
A graph is the skeleton graph of a simple orthogonal polyhedron ifand only if
I it is bipartite,
I it is planar,
I it is 3-regular,
I removal of any 2 of its vertices disconnects it into at most 2components.
Note: This gives a polynomial time characterization.
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 38: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/38.jpg)
Main result
A graph is the skeleton graph of a simple orthogonal polyhedron ifand only if
I it is bipartite,
I it is planar,
I it is 3-regular,
I removal of any 2 of its vertices disconnects it into at most 2components.
Note: This gives a polynomial time characterization.
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 39: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/39.jpg)
Main result
A graph is the skeleton graph of a simple orthogonal polyhedron ifand only if
I it is bipartite,
I it is planar,
I it is 3-regular,
I removal of any 2 of its vertices disconnects it into at most 2components.
Note: This gives a polynomial time characterization.
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 40: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/40.jpg)
Main result
A graph is the skeleton graph of a simple orthogonal polyhedron ifand only if
I it is bipartite,
I it is planar,
I it is 3-regular,
I removal of any 2 of its vertices disconnects it into at most 2components.
Note: This gives a polynomial time characterization.
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 41: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/41.jpg)
Main result
A graph is the skeleton graph of a simple orthogonal polyhedron ifand only if
I it is bipartite,
I it is planar,
I it is 3-regular,
I removal of any 2 of its vertices disconnects it into at most 2components.
Note: This gives a polynomial time characterization.
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
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Main result
A graph is the skeleton graph of a simple orthogonal polyhedron ifand only if
I it is bipartite,
I it is planar,
I it is 3-regular,
I removal of any 2 of its vertices disconnects it into at most 2components.
Note: This gives a polynomial time characterization.
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
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Proof sketch for xyz polyhedra
TheoremA graph is the skeleton graph of an xyz polyhedron if and only if itis
I planar
I bipartite
I 3-regular
I 3-connected
I Forward direction (polyhedron to graph) was proved in aprevious paper.
I For the other direction (graph to polyhedron), look at thedual graph.
I Because of 3-regularity, the dual will be a triangulation (i.e.each face will be bounded by exactly three edges).
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
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Proof sketch for xyz polyhedra
TheoremA graph is the skeleton graph of an xyz polyhedron if and only if itis
I planar
I bipartite
I 3-regular
I 3-connected
I Forward direction (polyhedron to graph) was proved in aprevious paper.
I For the other direction (graph to polyhedron), look at thedual graph.
I Because of 3-regularity, the dual will be a triangulation (i.e.each face will be bounded by exactly three edges).
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 45: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/45.jpg)
Proof sketch for xyz polyhedra
TheoremA graph is the skeleton graph of an xyz polyhedron if and only if itis
I planar
I bipartite
I 3-regular
I 3-connected
I Forward direction (polyhedron to graph) was proved in aprevious paper.
I For the other direction (graph to polyhedron), look at thedual graph.
I Because of 3-regularity, the dual will be a triangulation (i.e.each face will be bounded by exactly three edges).
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 46: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/46.jpg)
Proof sketch for xyz polyhedra
TheoremA graph is the skeleton graph of an xyz polyhedron if and only if itis
I planar
I bipartite
I 3-regular
I 3-connected
I Forward direction (polyhedron to graph) was proved in aprevious paper.
I For the other direction (graph to polyhedron), look at thedual graph.
I Because of 3-regularity, the dual will be a triangulation (i.e.each face will be bounded by exactly three edges).
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 47: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/47.jpg)
Corner polyhedra
TheoremA graph is the skeleton graph of a corner polyhedron if
I it is planar
I it is bipartite
I it is 3-regular
I its dual is 4-connected
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
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Corner polyhedra
TheoremA graph is the skeleton graph of a corner polyhedron if
I it is planar
I it is bipartite
I it is 3-regular
I its dual is 4-connected
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
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Corner polyhedra
TheoremA graph is the skeleton graph of a corner polyhedron if
I it is planar
I it is bipartite
I it is 3-regular
I its dual is 4-connected
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
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Corner polyhedra
TheoremA graph is the skeleton graph of a corner polyhedron if
I it is planar
I it is bipartite
I it is 3-regular
I its dual is 4-connected
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
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Corner polyhedra
TheoremA graph is the skeleton graph of a corner polyhedron if
I it is planar
I it is bipartite
I it is 3-regular
I its dual is 4-connected
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
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Corner polyhedra
A corner polyhedron is a simple orthogonal polyhedron for whichall but three faces are oriented towards the vector (1, 1, 1)
Figure: A corner polyhedron.
Note that a corner polyhedron is always xyz.
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
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Proof sketch for xyz polyhedra
TheoremIf a graph is planar, bipartite, 3-regular and 3-connected, then itcan be represented as the skeleton of an xyz polyhedron.
I Check: Is its dual 4-connected? If yes, then we are done.
I If no, then split the dual along a separating triangle.
Figure: Splitting along separting triangles.
I Pick the one for which one of the components has no moreseparating triangles.
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 54: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/54.jpg)
Proof sketch for xyz polyhedra
TheoremIf a graph is planar, bipartite, 3-regular and 3-connected, then itcan be represented as the skeleton of an xyz polyhedron.
I Check: Is its dual 4-connected? If yes, then we are done.
I If no, then split the dual along a separating triangle.
Figure: Splitting along separting triangles.
I Pick the one for which one of the components has no moreseparating triangles.
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 55: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/55.jpg)
Proof sketch for xyz polyhedra
TheoremIf a graph is planar, bipartite, 3-regular and 3-connected, then itcan be represented as the skeleton of an xyz polyhedron.
I Check: Is its dual 4-connected? If yes, then we are done.
I If no, then split the dual along a separating triangle.
Figure: Splitting along separting triangles.
I Pick the one for which one of the components has no moreseparating triangles.
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 56: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/56.jpg)
Proof sketch for xyz polyhedra
I The component with no separating triangle is 4-connected.
I So it gives a corner polyhedron.
I The other component is an xyz polyhedron by induction.
I So glue them together.
Figure: Gluing two polyhedra together.
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 57: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/57.jpg)
Proof sketch for xyz polyhedra
I The component with no separating triangle is 4-connected.
I So it gives a corner polyhedron.
I The other component is an xyz polyhedron by induction.
I So glue them together.
Figure: Gluing two polyhedra together.
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 58: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/58.jpg)
Proof sketch for xyz polyhedra
I The component with no separating triangle is 4-connected.
I So it gives a corner polyhedron.
I The other component is an xyz polyhedron by induction.
I So glue them together.
Figure: Gluing two polyhedra together.
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 59: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/59.jpg)
Summary of the proof
I Prove that a planar, bipartite, 3-regular graph whose dual is4-connected can be represented as a corner polyhedron, whichis always an xyz polyhedron.
I Use that and induction to give a characterization of xyzpolyhedra.
I Use the characterization of xyz polyhedra to give acharacterization of simple polyhedra. (Skipped. Usestechniques such as SPQR trees.)
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
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Summary of the proof
I Prove that a planar, bipartite, 3-regular graph whose dual is4-connected can be represented as a corner polyhedron, whichis always an xyz polyhedron.
I Use that and induction to give a characterization of xyzpolyhedra.
I Use the characterization of xyz polyhedra to give acharacterization of simple polyhedra. (Skipped. Usestechniques such as SPQR trees.)
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra
![Page 61: ``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein …cs763/Projects/vinayak.pdf · 2010. 7. 27. · Skeleton graphs of polytopes I Imagine a polytope. I \Polytope is](https://reader035.vdocuments.net/reader035/viewer/2022070211/60fcff6fa0fadf6fee42ae0a/html5/thumbnails/61.jpg)
Summary of the proof
I Prove that a planar, bipartite, 3-regular graph whose dual is4-connected can be represented as a corner polyhedron, whichis always an xyz polyhedron.
I Use that and induction to give a characterization of xyzpolyhedra.
I Use the characterization of xyz polyhedra to give acharacterization of simple polyhedra. (Skipped. Usestechniques such as SPQR trees.)
Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra