MAT 2720
Discrete Mathematics
Section 8.7
Planar Graphs
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Goals


Define Planar Graphs
The conditions for a graph to be
planar
• Series Reductions
• Homeomorphic Graphs
Example 1
The following are 2 ways of drawing the
same graph, K4.
Definition
A graph is planar if it can be drawn in
the plane without its edges crossing.
Definition
A graph is planar if it can be drawn in
the plane without its edges crossing.



K4 is planar
K5 is NOT planar
K3,3 is NOT planar
Faces of a Planar Graph
Euler’s Formula for Graphs
If G is a connected, planar graph with e
edges, v vertices, and f faces, then
f=e-v+2
Euler’s Formula for Graphs
If G is a connected, planar graph with e
edges, v vertices, and f faces, then
f=e-v+2
Example 2
K3,3 is NOT planar
Example 2
Suppose K3,3 is planar
1. Every cycle has at least 4
edges.
Example 2
Suppose K3,3 is planar
1. Every cycle has at least 4
edges.
2.The no. of edges that
bound faces is at least 4f
(with some edges counted
twice).
f=e-v+2
Observations
A graph contains K3,3 or K5 as a subgraph
is NOT planar.
Observations
A graph contains a graph “somewhat”
similar to K3,3 or K5 as a subgraph is
NOT planar.
Definitions (simplified)
c
Edges in Series
b
a
Series Reduction
c
c
b
a
a
Homeomorphic
Two graphs are homeomorphic if they can
be reduced to isomorphic graphs by a
sequence of series reduction.
Example 3
The following graphs are homeomorphic.
a
b
c
d
Finally…Kuratowski’s Theorem
A graph is planar iff it does not contain a
subgraph homeomorphic to K3,3 or K5 .
Example 3
Show that the following graph is not
planar.
Example 3
Key: Locate the subgraph homeomorphic
to K3,3 or K5
Example 3: Formal Solutions
Eliminating
edges (a,b),
(f,e), and (g,h)
eliminating
vertices g and h
Example 3: Formal Solutions
Eliminating
edges (a,b),
(f,e), and (g,h)
eliminating
vertices g and h
Since the graph contains a subgraph
homeomorphic to K3,3, it is not planar