Transcript Powerpoint

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.
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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.
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Which graphs are isomorphic to graph B?
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Petersen
Graph
?
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Planar graphs: Isomorphic graphs can have
planar embeddings that are not isomorphic.
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Automorphism: Isomorphism from an object
to itself. How many automorphisms does this
embedding have?
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The identity
automorphism.
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Flip over a horizontal axis.
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Rotate 180º
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Flip over the
vertical axis.
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Rotate 180º
Then flip over a horizonal axis.
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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)
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Two line notation?
Cycle structure
notation?
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Two line notation:
0 1 2 3 4 5
0 1 5 4 3 2
Cycle structure
notation:
(0)(1) (25)(34)
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Two line notation?
Cycle structure
notation?
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Two line notation:
0 1 2 3 4 5
1 0 4 5 2 3
Cycle structure
notation:
(01) (24)(35)
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Two line notation?
Cycle structure
notation?
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Two line notation:
0 1 2 3 4 5
1 0 3 2 4 4
Cycle structure
notation:
(01) (23)(45)
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Permutations that are automorphisms:
identity
(0)(1) (2)(3)(4)(5)
rotation
(01) (24)(35)
horizontal
flip
(0)(1)(25)(34)
vertical
flip
(01) (23)(45)
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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:
rotate 180º
(01) (24)(35) *
horizonal flip
(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
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identity
rotation
vertical
flip
Then flip over a horizonal axis.
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Two graphs that are isomorphic but their
embeddings are not:
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Flipping this over:
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If an embedding
has an
automorphism to
its flip then the
embedding is
not chiral.
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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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r
f
d
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