1. Realizability of Graphs Maria Belk and Robert Connelly

2. Graph Graphs: A graph has vertices…      

3. Graph Graphs: A graph has vertices and edges.      

4. Graph Graphs: A graph contains vertices and edges. Each edge connects two vertices.       The edge 

5. Realization Realization: A realization of a graph  is a placement of the vertices in some  .            

6. Realization Here are two realizations of the same graph:         

7.  -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.

8. Which graphs are  -realizable?

9. Tree: A connected graph without any cycles. Every tree is  -realizable.

10. Which graphs are  -realizable? The triangle is not  -realizable. But it is  -realizable.

11. Which graphs are  -realizable? The  -gon is not  -realizable. Neither is any graph that contains the  -gon.

12. Which graphs are  -realizable? The  -gon is not  -realizable. Neither is any graph that contains the  -gon.     

13. Theorem. (Connelly)  -realizable = Trees

14. Which graphs are  -realizable?

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

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

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

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

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

20. 2-tree: Start with a triangle. Attach another triangle along an edge. Continue attaching triangles to edges.  -realizability

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

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

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

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

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

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

27.  -trees are  -realizable.  -realizability

28. Partial  -tree: Subgraph of a  -tree

29.  -realizability

30. Partial  -tree: Subgraph of a  -tree  -realizability

31. Partial  -trees are also  -realizable.

32.  -realizability The tetrahedron is not  -realizable. But it is  -realizable.

33.  -realizability Theorem. (Belk and Connelly) The following are equivalent:  is a partial  -tree.  does not “contain” the tetrahedron.  is  -realizable.

34. Realizability AllowedForbidden  -realizabilityTrees  -realizabilityPartial  -trees

35. Which graphs are  -realizable?

36. 3-realizability  -tree: Start with a tetrahedron. Attach another tetrahedron along a triangle. Continue attaching tetrahedron to triangles.

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

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

39.  -realizability  -trees are  -realizable.

40.  -realizability Partial  -tree: Subgraph of a  -tree Partial 3-trees are 3-realizable.

41.  -realizability Partial  -tree: Subgraph of a  -tree Partial 3-trees are 3-realizable.

42.  -realizability Partial 3-tree: Subgraph of a  -tree Another example:

43.  -realizability Partial 3-tree: Subgraph of a  -tree Another example:

44.  -realizability   

45.    Not  -realizable Not  -realizable Not  -realizable

46.  -realizability Are the following all equal? Partial  -trees Not containing    -realizability

47. Are the following all equal? Partial  -trees Not containing    -realizability Answer: No, none of the three are equal.

48.  -realizability None of the reverse directions are true. Partial  -trees  -realizability Does not contain    

49. From Graph Theory: The following graphs are the “minimal” graphs that are not partial  -trees.  octahedron     

50. Which of these graphs is  -realizable?  octahedron     

51. Which of these graphs is  -realizable?  octahedron      NO Yes YES

52. Conclusion AllowedForbidden  -realizableTrees  -realizablePartial  -trees  -realizable Partial  -trees