Realizability of Graphs

Maria Belk and Robert Connelly

Graph

Graphs: A graph has vertices…

Graph

Graphs: A graph has vertices and edges.

Graph

Graphs: A graph contains vertices and edges.

Each edge connects two vertices.

The edge

Realization

Realization: A realization of a graph is a placement of the vertices in some .

Realization

Here are two realizations of the same graph:

-realizability

-realizable: A graph is -realizable if any realization can be moved into a -dimensional subspace without changing the edge lengths.

Example: A path is -realizable.

Which graphs are -realizable?

Which graphs are -realizable?

Tree: A connected graph without any cycles.

Every tree is -realizable.

Which graphs are -realizable?

The triangle is not -realizable.

But it is -realizable.

Which graphs are -realizable?

The -gon is not -realizable.

Neither is any graph that contains the -gon.

Which graphs are -realizable?

The -gon is not -realizable.

Neither is any graph that contains the -gon.

Theorem. (Connelly) -realizable = Trees

Which graphs are -realizable?

-tree: • Start with a triangle.• Attach another triangle along an edge.• Continue attaching triangles to edges.

-realizability

-tree: • Start with a triangle.• Attach another triangle along an edge.• Continue attaching triangles to edges.

-realizability

-tree: • Start with a triangle.• Attach another triangle along an edge.• Continue attaching triangles to edges.

-realizability

-tree: • Start with a triangle.• Attach another triangle along an edge.• Continue attaching triangles to edges.

-realizability

-tree: • Start with a triangle.• Attach another triangle along an edge.• Continue attaching triangles to edges.

-realizability

2-tree: • Start with a triangle.• Attach another triangle along an edge.• Continue attaching triangles to edges.

-realizability

-tree: • Start with a triangle.• Attach another triangle along an edge.• Continue attaching triangles to edges.

-realizability

-tree: • Start with a triangle.• Attach another triangle along an edge.• Continue attaching triangles to edges.

-realizability

-tree: • Start with a triangle.• Attach another triangle along an edge.• Continue attaching triangles to edges.

-realizability

-tree: • Start with a triangle.• Attach another triangle along an edge.• Continue attaching triangles to edges.

-realizability

-tree: • Start with a triangle.• Attach another triangle along an edge.• Continue attaching triangles to edges.

-realizability

-tree: • Start with a triangle.• Attach another triangle along an edge.• Continue attaching triangles to edges.

-realizability

-trees are -realizable.

-realizability

-realizability

Partial -tree: Subgraph of a -tree

Partial -tree: Subgraph of a -tree

-realizability

Partial -tree: Subgraph of a -tree

-realizability

-realizability

Partial -trees are also -realizable.

-realizability

The tetrahedron is not -realizable.

But it is -realizable.

-realizability

Theorem. (Belk and Connelly) The following are equivalent:

is a partial -tree. does not “contain” the tetrahedron. is -realizable.

Realizability

Allowed Forbidden

-realizability Trees

-realizability Partial -trees

Which graphs are -realizable?

3-realizability

-tree:• Start with a tetrahedron.• Attach another tetrahedron along a triangle.• Continue attaching tetrahedron to triangles.

-realizability

-tree:• Start with a tetrahedron.• Attach another tetrahedron along a triangle.• Continue attaching tetrahedra along triangles.

-realizability

-tree:• Start with a tetrahedron.• Attach another tetrahedron along a triangle.• Continue attaching tetrahedron to triangles.

-realizability

-trees are -realizable.

-realizability

Partial -tree: Subgraph of a -tree

Partial 3-trees are 3-realizable.

-realizability

Partial -tree: Subgraph of a -tree

Partial 3-trees are 3-realizable.

-realizability

Partial 3-tree: Subgraph of a -tree

Another example:

-realizability

Partial 3-tree: Subgraph of a -tree

Another example:

-realizability

-realizability

Not -realizable Not -realizableNot -realizable

-realizability

Are the following all equal?

• Partial -trees

• Not containing

-realizability

-realizability

Are the following all equal?

• Partial -trees

• Not containing

-realizability

Answer: No, none of the three are equal.

-realizability

None of the reverse directions are true.

Partial -trees

-realizability Does not contain

From Graph Theory: The following graphs are the “minimal” graphs that are not partial -trees.

octahedron

Which of these graphs is -realizable?

octahedron

Which of these graphs is -realizable?

octahedron

NO NO

Yes YES

ConclusionAllowed Forbidden

-realizable Trees

-realizable Partial -trees

-realizable

Partial -trees