Graph Isomorphism

1. Give the parent, queue, BFI (breadth first index), and level arrays when BFS is applied to this graph starting at vertex 0. Process the neighbours of each vertex in numerical order.

2. Graph Isomorphism The graph isomorphism problem has no known polynomial time algorithm which works for an arbitrary graph. Canonical form: If two graphs are isomorphic, their canonical forms must be the same, otherwise, they must be different. For trees and planar graphs, a canonical form can be computed in polynomial time.

3. Which graphs are isomorphic to graph B? Petersen Graph ? ? ?

7. Planar graphs: Isomorphic graphs can have planar embeddings that are not isomorphic.

8. Automorphism: Isomorphism from an object to itself. How many automorphisms does this embedding have?

10. The identity automorphism.

11. Flip over a horizontal axis.

12. Rotate 180º

13. Flip over the vertical axis.

14. Rotate 180º Then flip over a horizonal axis.

15. The original embedding. Identity automorphism: Two line notation: 0 1 2 3 4 5 0 1 2 3 4 5 Cycle structure notation: (0) (1)(2)(3)(4)(5)

16. Two line notation? Cycle structure notation?

17. Two line notation: 0 1 2 3 4 5 0 1 5 4 3 2 Cycle structure notation: (0)(1) (25)(34)

18. Two line notation? Cycle structure notation?

19. Two line notation: 0 1 2 3 4 5 1 0 4 5 2 3 Cycle structure notation: (01) (24)(35)

20. Two line notation? Cycle structure notation?

21. Two line notation: 0 1 2 3 4 5 1 0 3 2 4 4 Cycle structure notation: (01) (23)(45)

22. Permutations that are automorphisms: horizontal flip identity (0)(1)(25)(34) (0)(1) (2)(3)(4)(5) rotation vertical flip (01) (23)(45) (01) (24)(35)

23. The automorphism form a group: • The identity is always included. • If p is an automorphism, then so is p-1. • If p and q are automorphisms, then so is p * q. • What is: • rotate 180º horizonal flip • (01) (24)(35) * (0)(1)(25)(34)

24. The automorphism form a group: • The identity is always included. • If p is an automorphism, then so is p-1. • If p and q are automorphisms, then so is p * q. • What is: • rotation horizonal flip • (01)(24)(35) * (0)(1)(25)(34) • = (01)(23)(45) vertical flip

25. identity vertical flip rotation Then flip over a horizonal axis.

26. Two graphs that are isomorphic but their embeddings are not:

28. Flipping this over:

29. If an embedding has an automorphism to its flip then the embedding is not chiral.

32. If an embedding has no automorphisms to its flip then the embedding is chiral. Chiral embeddings have a sense of clickwise.

33. Clockwise_BFS(r, f, d): • Choose a root vertex r. • Choose a first child vertex f. • Choose a direction d (clockwise or countercloswise) • Do BFS subject to: • The children of each vertex are • visited in the chosen order starting • with f for the root or otherwise, • starting with the BFS parent.

34. r d f