1. Realizability of Graphs Maria Belk and Robert Connelly

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

3. Motivation: the molecule problem 1.Is this a possible configuration of points in 3 dimensions? 2.What is a possible configuration satisfying these distances? 7 7 4 4 7 4 2 5

4. Motivation: the molecule problem 1.Is this a possible configuration of points in 3 dimensions? 2.What is a possible configuration satisfying these distances? Unfortunately, these questions are NP-hard. We will answer a different question.

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. 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 11 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 it does not have any   as a minor.

15. Forbidden Minors 1-realizable  2-realizable   + ? 3-realizable   + ?

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 1-realizable  2-realizable  3-realizable   + ?

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?   NO       

27. Which of these graph is 3-realizable?   NO   NO     

28. Which of these graph is 3-realizable?   NO   NO   YES     YES

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? 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  . NO

31. Forbidden Minors 1-realizable  2-realizable  3-realizable   +  

32.   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 Forbidden Minors Building Blocks 1-realizable   2-realizable   3-realizable  ,    ,  ,    

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.