Steinitzs Theorem 1 Skeletons of Polyhedra Jon Hillery

Steinitz’s Theorem: 1 -Skeletons of Polyhedra Jon Hillery

Overview ● We can completely classify which graphs can come from convex polyhedra ● This allows us to find interesting ways to represent certain graphs

Convexity Definition: a region of space is convex if any line you draw between points in the region stays inside the region.

Convexity Convex Not Convex

Convexity Convex Not Convex

Convexity Convex Not Convex

What is a 1 -Skeleton? Take a convex polyhedron and look only at the graph formed from its vertices and edges. This is the polyhedron’s 1 -Skeleton

What is a 1 -Skeleton?

What is a 1 -Skeleton?

What properties must the resulting graph have?

The graph must be planar!

The Graph Must be Planar! General Strategy: 1. Move your eye very close to a face until the edges of that face appear outside all others in you field of view

The graph must be planar!

The Graph Must be Planar! General Strategy: 2. Because the polyhedron is convex, you will see all of the edges unobstructed - a planar graph!

What else can we say?

3 -Connected Graphs Definition: A 3 -connected graph : 1. Has more than 3 vertices 2. Stays connected if any two vertices are removed.

1. More than 3 Vertices

1. More than 3 Vertices

1. More than 3 Vertices We have to have these outside edges

1. More than 3 Vertices And all of this space

1. More than 3 Vertices If we added more than that it wouldn’t be convex anymore! Therefore, we need at least four vertices for our shape to be three-dimensional.

1. More than 3 Vertices (Side note: this process is called a convex hull , and you can use linear algebra to show that the convex hull of 3 points must be in a plane and therefore 2 dimensional. )

2. Stays connected if any two vertices are removed.

2. Stays connected if any two vertices are removed.

Are these “all” of the properties of a 1 Skeleton?

Steinitz’s Theorem : Any graph which is planar and 3 -connected is the 1 -Skeleton of some polyhedron.

Applications

Straight Line Drawings Theorem: Any planar graph can be drawn with straight lines!

Straight Line Drawings Proof Idea: 1. Take a planar graph 2. Add enough vertices and edges to make it 3 -connected 3. Look at the polyhedron. 4. Remove the extra added edges.

Straight Line Drawings

Straight Line Drawings It is still planar

Straight Line Drawings It’s a pyramid!

Straight Line Drawings It’s a pyramid!

Straight Line Drawings

Straight Line Drawings A Remove the edges we added B B C D C A E D E

Circle-Packing/Spherical Representations If a graph is 3 connected and planar, we can represent it as a network of tangent circles where the centers of the circles are vertices and their tangencies are edges.

Circle-Packing/Spherical Representations To do this, take the midsphere of the polyhedron (a sphere tangent to every edge).

Conclusion

Summary ● Every 1 -Skeleton of a polyhedron is planar and 3 -connected ● In fact, every 3 -connected planar graph is the 1 Skeleton of some polyhedron ● We can use this fact to find nice representations (tangent circles and straight-line drawings) of graphs

References ● ● ● ● https: //www. math. ucla. edu/~pak/geompol 8. pdf https: //link. springer. com/book/10. 1007%2 F 978 -1 -4613 -8431 -1 https: //www. mathsisfun. com/geometry/tetrahedron. html http: //bit-player. org/2012/dancing-with-the-spheres https: //en. wikipedia. org/wiki/File: Icosahedron-wireframe. jpg https: //www. crystalhealingart. nl/shop/dodecaeder-24 k-verguld/ https: //thenounproject. com/term/square-pyramid/405706/ https: //en. wikipedia. org/wiki/Midsphere#/media/File: Midsphere. png