Realizability of Graphs Maria Belk and Robert Connelly
Realizability of Graphs Maria Belk and Robert Connelly
Graphs: A graph has vertices…
Graphs: A graph has vertices and edges.
Graphs: A graph contains vertices and edges. Each edge connects two vertices. The edge
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?
-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 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 Partial -tree: Subgraph of a -tree
-realizability Partial -tree: Subgraph of a -tree
-realizability Partial -tree: Subgraph of a -tree
-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 Trees realizability Partial realizability trees Forbidden
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
-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 Partial -trees Does not -realizability contain None of the reverse directions are true.
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? NO Yes octahedron NO YES
Conclusion -realizable Allowed Trees -realizable Partial -trees -realizable Forbidden
- Slides: 52
