``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

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“Steinitz Theorems for Orthogonal Polyhedra” byDavid Eppstein and Elena Mumford (SoCG 2010)

Vinayak Pathak

July 25, 2010

Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra

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.













Figure: The skeleton of an Icosidodecahedron



Figure: The skeleton of a four dimensional cube



Figure: The skeleton of a complicated four dimensional polytope



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?



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).












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.









I A graph is a skeleton graph of a polyhedron if and only if. . .

no one knows.













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?


Convex polyhedra

I Steinitz Theorem!

A graph is the skeleton graph of a convex polyhedron if andonly if it’s 3-connected and planar.





Convex polyhedra






Convex polyhedra






Convex polyhedra






Simple orthogonal polyhedra

I “Orthogonal” means each face is perpendicular to one of thecoordinate axes.

I “Simple” means three things.











It should have the topology of a sphere.

Figure: Not allowed.



Exactly three mutually perpendicular edges should meet at eachvertex.




Each face should be simply connected.




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.)


Observations


The skeleton must be

I planar

I bipartite

I 3-regular

I 2-connected


Observations



I planar

I bipartite

I 3-regular

I 2-connected


Observations



I planar

I bipartite

I 3-regular

I 2-connected


Observations



I planar

I bipartite

I 3-regular

I 2-connected


Does the converse hold?

Figure: A planar, bipartite, 3-regular and 2-connected graph that is notthe skeleton of a simple orthogonal polyhedron.


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


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.


xyz polyhedra

Unfortunately, there are simple orthogonal polyhedra that are notxyz.

Figure: Not xyz.


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.


Main result


I it is bipartite,

I it is planar,

I it is 3-regular,




Main result


I it is bipartite,

I it is planar,

I it is 3-regular,




Main result


I it is bipartite,

I it is planar,

I it is 3-regular,




Main result


I it is bipartite,

I it is planar,

I it is 3-regular,




Main result


I it is bipartite,

I it is planar,

I it is 3-regular,




Proof sketch for xyz polyhedra


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).




I planar

I bipartite

I 3-regular

I 3-connected







I planar

I bipartite

I 3-regular

I 3-connected







I planar

I bipartite

I 3-regular

I 3-connected





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


Corner polyhedra


I it is planar

I it is bipartite

I it is 3-regular



Corner polyhedra


I it is planar

I it is bipartite

I it is 3-regular



Corner polyhedra


I it is planar

I it is bipartite

I it is 3-regular



Corner polyhedra


I it is planar

I it is bipartite

I it is 3-regular



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.



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.

















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.
















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.)












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``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