1. MAT 2720 Discrete Mathematics
Section 8.7
Planar Graphs

2. Minimum to copy for this section...
Proofs with graphical components

3. Goals Define Planar Graphs The conditions for a graph to be planar
Series Reductions
Homeomorphic Graphs

4. Example 1
The following are 2 ways of drawing the same graph, K4.

5. Definition
A graph is planar if it can be drawn in the plane without its edges crossing.

6. 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

7. 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 We will look at why.

8. Faces of a Planar Graph

9. 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

10. 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

11. Example 2
K3,3 is NOT planar

12. Example 2: Proof by Contradiction
Suppose K3,3 is planar
1. Every cycle has at least 4 edges.

13. Example 2: Proof by Contradiction
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

14. Observations
A graph contains K3,3 or K5 as a subgraph is NOT planar.

15. Eliminating edges (𝑎, 𝑏), (𝑏, 𝑐), and (𝑐, 𝑎)
Formal Solutions
Eliminating edges (𝑎, 𝑏), (𝑏, 𝑐), and (𝑐, 𝑎)
Since the graph contains a subgraph of 𝐾3,3, it is not planar.

16. Observations
A graph contains a graph “somewhat” similar to K3,3 or K5 as a subgraph is NOT planar.

17. Series Reduction
Edges in Series Series Reduction

18. Homeomorphic
Two graphs are homeomorphic if they can be reduced to isomorphic graphs by a sequence of series reduction.

19. Example 3
Show that the following graphs are homeomorphic.

20. Formal Solutions
The graphs are homeomorphic since they can be reduced to the same graph by a sequence of series reduction.
Series Reduction: eliminating vertices 𝑎 and 𝑏
Series Reduction: eliminating vertices 𝑐 and 𝑑

21. Back to our Earlier Example...

22. Formal Solutions
Series Reduction: eliminating vertex 𝑑
Eliminating edges (𝑎, 𝑏), (𝑏, 𝑐), and (𝑐, 𝑎)
Since the graph contains a subgraph homeomorphic to 𝐾3,3, it is not planar

23. Kuratowski’s Theorem
A graph is planar iff it does not contain a subgraph homeomorphic to 𝐾3,3 or 𝐾5 .

24. Example 3
Show that the following graph is not planar.

25. Example 3
Key: Locate the subgraph homeomorphic to K3,3 or K5

26. Example 3: Formal Solutions
Eliminating edges (a,b), (f,e), and (g,h)
eliminating vertices g and h

27. 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