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.
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. 2
8
Automorphism: Isomorphism from an object to itself. How many automorphisms does this embedding have?
15
The original embedding.
Identity automorphism:Two line notation:0 1 2 3 4 50 1 2 3 4 5
Cycle structure notation:
(0) (1)(2)(3)(4)(5)
22
(0)(1)(25)(34)(0)(1) (2)(3)(4)(5)
(01) (24)(35) (01) (23)(45)
Permutations that are automorphisms:
identity
rotation
horizontal flip
vertical flip
23
The automorphism form a group:1.The identity is always included.2.If p is an automorphism, then so is p-
1.3.If p and q are automorphisms, then so is p * q.
What is:rotate 180º horizonal flip(01) (24)(35) * (0)(1)(25)(34)
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The automorphism form a group:1.The identity is always included.2.If p is an automorphism, then so is p-
1.3.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
32
If an embedding has no automorphisms to its flip then the embedding is chiral.
Chiral embeddings have a sense of clickwise.
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Clockwise_BFS(r, f, d):1.Choose a root vertex r.2.Choose a first child vertex f.3.Choose a direction d (clockwise or countercloswise)4.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.
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