Realizability of Graphs

1. Realizability of Graphs Maria Belk and Robert Connelly

2. A chemist determines the distances between atoms in a molecule: 2 4 7 7 4 5 4 7 Motivation: the molecule problem

3. Is this a possible configuration of points in 3 dimensions? What is a possible configuration satisfying these distances? 2 4 7 7 4 5 4 7 Motivation: the molecule problem

4. Is this a possible configuration of points in 3 dimensions? What is a possible configuration satisfying these distances? Unfortunately, these questions are NP-hard. We will answer a different question. Motivation: the molecule problem

5. Definitions Definition: A realization of a graph is a function which assigns to each vertex of  a point  in some Euclidean space. Definition: A graph is -realizable if given any realization , …,  of the graph in some finite dimensional Euclidean space, there exists a realization , …,  in -dimensional Euclidean space with the same edge lengths.

6. Definition: A graph is -realizable if given any realization , …,  of the graph in some finite dimensional Euclidean space, there exists a realization , …,  in -dimensional Euclidean space with the same edge lengths. Example: A path is 1-realizable.

7. Examples • A tree is 1-realizable. •  is 2-realizable, but not 1-realizable. • A cycle is also 2-realizable, but not 1-realizable. • Any graph containing a cycle is not 1-realizable.

8. Examples • A tree is 1-realizable. •  is 2-realizable, but not 1-realizable. • A cycle is also 2-realizable, but not 1-realizable. • Any graph containing a cycle is not 1-realizable.

9. Examples • A tree is 1-realizable. •  is 2-realizable, but not 1-realizable. • A cycle is also 2-realizable, but not 1-realizable. • Any graph containing a cycle is not 1-realizable. 1 1 1 0 0

10. Examples • A tree is 1-realizable. •  is 2-realizable, but not 1-realizable. • A cycle is also 2-realizable, but not 1-realizable. • Any graph containing a cycle is not 1-realizable.

11. Theorem (Connelly) A graph is 1-realizable if and only if it is a forest (a disjoint collection of trees).

12. Definition. A minor of a graph is a graph obtained by a sequence of • Edge deletions and • Edge contractions (identify the two vertices belonging to the edge and remove any loops or multiple edges). Theorem. (Connelly) If is -realizable then every minor of  is -realizable (this means -realizability is a minor monotone graph property).

13. Graph Minor Theorem Theorem (Robertson and Seymour). For a minor monotone graph property, there exists a finite list of graphs , …,  such that a graph satisfies the minor monotone graph property if and only if does not have as a minor. Example: A graph is 1-realizable if and only if it does not contain  as a minor.

14. Graph Minor Theorem By the Graph Minor Theorem: For each , there exists a finite list of graphs , …,  such that a graph is -realizable if and only if itdoes not have any  as a minor.

15. Forbidden Minors

16. -sum Suppose  and  both contain a . The -sum of  and  is the graph obtained by identifying the two ’s.  

17. -trees -tree: -sums of ’s Example: a 1-tree is a tree

18. -trees -tree: -sums of ’s Example: 2-tree

19. -trees -tree: -sums of ’s Example: 3-tree

20. Partial -trees Partial -tree: subgraph of a -tree Example: partial 2-tree

21. Partial k-trees Theorem (Connelly). Partial -trees are -realizable.

22. 2-realizability Theorem (Wagner, 1937). is a partial 2-tree if and only if does not contain  as a minor. Theorem (Belk, Connelly).  is 2-realizable if and only if  does not contain  as a minor.

23. Forbidden Minors

24. Theorem (Arnborg, Proskurowski, Corneil) The forbidden minors for partial 3-trees.

25. Which of these graphs is 3-realizable?

26. Which of these graph is 3-realizable?

27. Which of these graph is 3-realizable?

28. Which of these graph is 3-realizable?

29. Are there any more forbidden minors for 3-realizability? Lemma (Belk and Connelly). If contains  or  as a minor then either • contains  or  as a minor, or • can be constructed by 2-sums and 1-sums of partial 3-trees,’s, and ’s (and is thus 3-realizable). Theorem (Belk and Connelly). The forbidden minors for 3-realizability are  and .

30. Are there any more forbidden minors for 3-realizability? NO Lemma (Belk and Connelly). If contains  or  as a minor then either • contains  or  as a minor, or • can be constructed by 2-sums and 1-sums of partial 3-trees,’s, and ’s (and is thus 3-realizable). Theorem (Belk and Connelly). The forbidden minors for 3-realizability are  and .

31. Forbidden Minors

32.  is not 3-realizable

33. is not 3-realizable

34.  is not 3-realizable

35.  is not 3-realizable

36.  is not 3-realizable

37.  is not 3-realizable

38.  is not 3-realizable

39.  is not 3-realizable

40.  and  Theorem (Belk).  and  are 3-realizable. Idea of Proof: Pull 2 vertices as far apart as possible.

41. Conclusion

42. Open Question • Which graphs are 4-realizable? •  is a forbidden minor. • There is a generalization of, using a tetrahedron and a degenerate triangle. • However, there are over 75 forbidden minors for partial 4-trees.