MAT 2720Discrete 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)

Edges in Series

Series Reductiona

c

b

a

c

b

a

c

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