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1/25/2005 Tucker, Sec. 1.2 1
Applied Combinatorics, 4th Ed.Alan Tucker
Section 1.2
Isomorphism
Prepared by Jo Ellis-Monaghan
1/25/2005 Tucker, Sec. 1.2 2
• Two graphs G and are isomorphic if :
– There exists a one-to-one correspondence between vertices in G and , such that
– There is an edge between a and b in G if and only if there is an edge between the corresponding vertices and in .
• The definition for oriented graphs is the same, except the head and tail of each edge of
G must correspond to the head and tail in .
Definition of Isomorphism
G
G
G
G
1/25/2005 Tucker, Sec. 1.2 3
Example of isomorphic graphs
G
a b
cd
ef
1 23
4
56
G
An isomorphism between G and :
a 6 d 5
b 1 e 2
c 3 f 4
G
1/25/2005 Tucker, Sec. 1.2 4
Kn, the complete graph on n vertices
K2
K3
K5K8K6
K4
K1
1/25/2005 Tucker, Sec. 1.2 5
The complement of a graphThe complement of G has all the edges that are missing in G—
i.e. that would have to be added to make the complete graph.
G
K6
G
1/25/2005 Tucker, Sec. 1.2 6
Advantage of the complement
• Theorem:Two graphs, G and H, are isomorphic if and only if their complements are.
In practice this means that we work with whichever of G or has few edges.G
1/25/2005 Tucker, Sec. 1.2 7
Subgraphs• Definition: if G is a graph, a subgraph H of G consists of a
subset V of the vertices of G, and a subset of the edges of G with endpoints in V.
n
a b
c d e fg
h
i
j k
lm
o
A graph G
c d e
i j k
h
g
m
f
l
Two subgraphs of G
1/25/2005 Tucker, Sec. 1.2 8
Induced subgraphs
• Choose a subset of the vertices of G, then include only the edges among those vertices.
n
a b
c d e fg
h
i
j k
lm
o
A graph GSubgraph induced by the vertices of degree 4.
d e
hj k
d e
hj k
1/25/2005 Tucker, Sec. 1.2 9
Elementary properties of isomorphic graphs
• Edge and vertex counts– Isomorphic graphs have the same number of edges and
vertices.
• Vertex sequence (the list of vertex degrees)– Isomorphic graphs have the same vertex sequences.
• Warning!! These can be used to show two graphs are not isomorphic, but can not show that two graphs are isomorphic.
1/25/2005 Tucker, Sec. 1.2 10
Two non-isomorphic graphs
Vertices: 6
Edges: 7
Vertex sequence: 4, 3, 3, 2, 2, 0.
Vertices: 6
Edges: 7
Vertex sequence: 5, 3, 2, 2, 1, 1.
1/25/2005 Tucker, Sec. 1.2 11
Subgraph properties ofisomorphic graphs
• Isomorphic graphs have the same sets of subgraphs:– there is a one-to-one correspondence between the
subgraphs such that corresponding subgraphs are isomorphic.
• Typically check induced subgraphs, or number of a specific kind of subgraphs such as cycles or cliques.
• Warning!! These can be used to show two graphs are not isomorphic, but can not show that two graphs are isomorphic.
1/25/2005 Tucker, Sec. 1.2 12
Two non-isomorphic graphs
a
c
d
b
e
f g
h
2 3
5
6 7
8
1 4
Vertices: 8
Edges: 10
Vertex sequence: 3, 3, 3, 3, 2, 2, 2, 2.
Vertices: 8
Edges: 10
Vertex sequence: 3, 3, 3, 3, 2, 2, 2, 2.
However, induced subgraphs on degree 3 vertices are NOT isomorphic!
3
1/25/2005 Tucker, Sec. 1.2 13
An approach to checking isomorphism:
Count the vertices. The graphs must have an equal number.
Count the edges. The graphs must have an equal number.
Check vertex degree sequence. Each graph must have the same degree sequence.
Check induced subgraphs for isomorphism. If the subgraphs are not isomorphic, then the larger graphs are not either.
Count numbers of cycles/cliques.
If these tests don’t help, and you suspect the graphs actually are isomorphic, then try to find a one-to-one correspondence between vertices of one graph and vertices of the other. Remember that a vertex of degree n in the one graph must correspond to a vertex of degree n in the other.
1/25/2005 Tucker, Sec. 1.2 14
For the class to try:
a
b f
e
c d
1
2
3
4
5
6
Are these pairs of graphs isomorphic?
#1
#2
Isomorphic: a-1, b-5, c-4, d-3, e-2, f-6.
Not Isomorphic:5 K3’s on left,4 K3’s on right.
a b c
d
f ge
1 2
34
5 6
7
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