Planarity
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Overview
Planar Graphs
Euler’s Formula
Some Non-Planar Graphs
Causes of Non-Planarity
Kuratowski’s Theorem
Planarity Algorithm for Hamiltonian Graphs
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Outline
Planar Graphs
Euler’s Formula
Some Non-Planar Graphs
Causes of Non-Planarity
Kuratowski’s Theorem
Planarity Algorithm for Hamiltonian Graphs
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Planar Graphs
Definition
A graph is planar if it can be drawn in the plane with no edges
crossing.
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Examples of Planar Graphs
C4 and K4 are both planar.
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Outline
Planar Graphs
Euler’s Formula
Some Non-Planar Graphs
Causes of Non-Planarity
Kuratowski’s Theorem
Planarity Algorithm for Hamiltonian Graphs
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Euler’s Formula
Theorem
For any connected plane graph with p vertices and q edges
that divides the plane into r regions, we have
p − q + r = 2.
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Euler’s Formula
Theorem
For any connected plane graph with p vertices and q edges
that divides the plane into r regions, we have
p − q + r = 2.
Proof Idea
Remove edges until no cycles exist, always maintaining
connectivity.
Each removal of an edge from a cycle reduces the number of
edges by 1 and the number of regions by 1, and leaves the
number of vertices unchanged.
The resulting graph is a tree, which has one less edge than
vertices, and which has one region.
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Outline
Planar Graphs
Euler’s Formula
Some Non-Planar Graphs
Causes of Non-Planarity
Kuratowski’s Theorem
Planarity Algorithm for Hamiltonian Graphs
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Regional Degree Theorem
Definition
The degree of a region of a plane graph is the number of
encounters with edges in a walk around the boundary of the
region.
Theorem
For any connected planar graph with q edges that divides the
plane into r regions, we have
2q = sum of degrees of the regions .
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Example of a Non-Planar Graph
Consider a connected graph with p = 6 vertices and q = 13
edges.
If the graph were planar, then by Euler’s formula it would have
r = 9 regions.
c
b
a
f
d
e
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K5 is not planar
K5 has p = 5 vertices and q = 10 edges.
If K5 were planar, it would have r = 7 regions.
b
a
c
e
d
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K3,3 is not planar
K3,3 has p = 6 vertices and q = 9 edges.
If K3,3 were planar, it would have r = 5 regions.
a
b
c
d
e
f
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Petersen graph is not planar
Petersen graph has p = 10 vertices and q = 15 edges.
If Petersen graph were planar, it would have r = 7 regions.
g
h
b
c
a
f
d
i
e
j
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Outline
Planar Graphs
Euler’s Formula
Some Non-Planar Graphs
Causes of Non-Planarity
Kuratowski’s Theorem
Planarity Algorithm for Hamiltonian Graphs
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What makes a graph non-planar?
I
Does having large cycles make a graph non-planar?
I
Does having many edges make a graph non-planar?
I
Does having a non-planar subgraph make a graph
non-planar?
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Planarity of Complete Graphs
Theorem
Kn is planar if and only if n ≤ 4.
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Planarity of Complete Bipartite Graphs
Theorem
Km,n is planar if and only if min(m, n) ≤ 2.
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Is this graph non-planar?
c
b
a
f
d
e
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Outline
Planar Graphs
Euler’s Formula
Some Non-Planar Graphs
Causes of Non-Planarity
Kuratowski’s Theorem
Planarity Algorithm for Hamiltonian Graphs
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Subdivision of graphs
Definition
Inserting a new vertex into an existing edge of a graph is
called subdividing the edge.
One or more subdivisions of edges creates a subdivision of the
original graph.
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Kuratowski’s Theorem
Theorem
A graph is planar if and only if it does not contain a
subdivision of K5 or K3,3 as a subgraph.
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Outline
Planar Graphs
Euler’s Formula
Some Non-Planar Graphs
Causes of Non-Planarity
Kuratowski’s Theorem
Planarity Algorithm for Hamiltonian Graphs
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Planarity Algorithm for Hamiltonian Graphs
Let G be a Hamiltonian graph.
1. Draw the graph G with Hamilton cycle H on the outside,
i.e. with H as the boundary of the infinite region.
2. List the edges of G not in H: e1 , . . . , er .
3. Form a new graph K in which the vertices are labelled
e1 , . . . , er and where the vertices labelled ei , ej are joined
by an edge if and only if ei , ej cross in the drawing of G ,
i.e. cannot both be drawn inside (or outside) H (such
edges are said to be incompatible).
4. Then G is planar if and only if K is bipartite.
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Example
Verify that the given graph has a Hamilton cycle and find a
plane diagram for the graph.
b
c
a
d
e
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Example
Hamilton cycle drawn on
the outside.
c
Corresponding incompatible
edge graph.
ab
ad
ce
bc
a
d
b
e
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Acknowledgements
Statements of results follow the notation and wording of
Anderson’s First Course in Discrete Mathematics.
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