``Steinitz Theorems for Orthogonal Polyhedra' by David Eppstein...

“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.

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.

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.

Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra

Skeleton graphs of polytopes

Figure: The skeleton of an Icosidodecahedron

Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra

Skeleton graphs of polytopes

Figure: The skeleton of a four dimensional cube

Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra

Skeleton graphs of polytopes

Figure: The skeleton of a complicated four dimensional polytope

Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Simple orthogonal polyhedra

It should have the topology of a sphere.

Figure: Not allowed.

Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra

Simple orthogonal polyhedra

Exactly three mutually perpendicular edges should meet at eachvertex.

Figure: Not allowed.

Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra

Simple orthogonal polyhedra

Each face should be simply connected.

Figure: Not allowed.

Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra

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

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

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

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

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

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

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

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

xyz polyhedra

Unfortunately, there are simple orthogonal polyhedra that are notxyz.

Figure: Not xyz.

Vinayak Pathak Steinitz Theorems for Orthogonal Polyhedra

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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