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Realizability of Graphs

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. .

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Realizability of Graphs

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  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. Which graphs are -realizable? 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. -realizability -tree: • Start with a triangle. • Attach another triangle along an edge. • Continue attaching triangles to edges.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

  31. -realizability 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

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

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

  47. -realizability 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. Does not contain  Partial -trees   -realizability

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

  50. Which of these graphs is -realizable?

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