Graph Isomorphism

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# Graph Isomorphism - PowerPoint PPT Presentation

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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## Graph Isomorphism

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Presentation Transcript

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

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.

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

The identity

automorphism.

Rotate 180º

Then flip over a horizonal axis.

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)

Two line notation?

Cycle structure notation?

Two line notation:

0 1 2 3 4 5

0 1 5 4 3 2

Cycle structure notation:

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

Two line notation?

Cycle structure notation?

Two line notation:

0 1 2 3 4 5

1 0 4 5 2 3

Cycle structure notation:

(01) (24)(35)

Two line notation?

Cycle structure notation?

Two line notation:

0 1 2 3 4 5

1 0 3 2 4 4

Cycle structure notation:

(01) (23)(45)

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)

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)

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

identity

vertical

flip

rotation

Then flip over a horizonal axis.

not chiral.

chiral.

Chiral embeddings have a sense of clickwise.

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.

r

d

f