4'5'1 Graph Isomorphism

1
4.5.1 Graph Isomorphism

• Isomorphism
• Isomorphic

2
4.5.2-4 Planar Graphs

• Planar graph, planar representation, face
• Euler's Theorem
• Girth, edge-vertex inequality

3
Theorem 4.5.1 Euler's Theorem

• For a planar connected graph
• V-EF 2
• Prove by induction on F .

4
Example 4.5.4 Euler's Theorem
Figure 4, Page 219 Graph G V9, E16,
F9, V-EF9-1692 Graph H V9, E17,
F10, V-EF9-17102 Graph I V9,
E8, F1, V-EF9-812
5
Theorem 4.5.1 Euler's Theorem
Proof  For every natural F define the statement 
  statement F   every planar connected graph
with F faces, V vertices, and E edges satisfies
V-EF2.   A graph with a cycle has at least two
faces one inside and one outside.  Therefore a
connected graph with just one face has no cycle,
and is a tree, having exactly one more vertex
than edge  V-E1.   This implies that 
(V-E)F112,  so statement 1  is true.
6
Theorem 4.5.1 Euler's Theorem
Now choose any natural number F for which
statement F  is true, and let G' be any planar
connected graph with F'F1 faces, V' vertices,
and E' edges.  Since G' has at least two faces,
G' has a cycle.  Remove one edge from that cycle,
joining two faces and giving a subgraph G with F
faces, V vertices, and E edges, where VV' and
E1E'.  Therefore        V' - E' F'      V -
(E1) (F1)      V - E F      2   and
statement F1  is a consequence of statement F. 
 Since F was arbitrary, then by the principle of
mathematical induction statement F  is true for
every F.
7
Start Lesson 21aQ20--
8
Definition 4.5.4 Girth
The girth g of a connected graph is the number of
edges in its shortest cycle.
9
Proof of Theorem 4.5.2
10
Concluded Proof of Theorem 4.5.2
11
Theorem 4.5.2 Edge-Vertex Inequality
This is an upper bound for the number of edges.
12
Example 4.5.5
Therefore the graph is NOT planar.
13
A
14
K5
15
B
16
K3,3
17
P
18
End Lesson 21a
19
(No Transcript)
20
MATHEMATICAL SYMBOLS

• ? ? ? ? ? ? ? ? ?
• ? ? ? ? ? ?
• ? ? ? ? ? ? ? ?
• ? ? ? ? ?