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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. Graph Isomorphism

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slide1

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.

slide2

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.

slide8

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

slide10

The identity

automorphism.

slide14

Rotate 180º

Then flip over a horizonal axis.

slide15

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)

slide16

Two line notation?

Cycle structure notation?

slide17

Two line notation:

0 1 2 3 4 5

0 1 5 4 3 2

Cycle structure notation:

(0)(1) (25)(34)

slide18

Two line notation?

Cycle structure notation?

slide19

Two line notation:

0 1 2 3 4 5

1 0 4 5 2 3

Cycle structure notation:

(01) (24)(35)

slide20

Two line notation?

Cycle structure notation?

slide21

Two line notation:

0 1 2 3 4 5

1 0 3 2 4 4

Cycle structure notation:

(01) (23)(45)

slide22

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)

slide23

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)
slide24

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
slide25

identity

vertical

flip

rotation

Then flip over a horizonal axis.

slide32

If an embedding has no automorphisms to its flip then the embedding is

chiral.

Chiral embeddings have a sense of clickwise.

slide33

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.
slide34

r

d

f