proof of Wagner’s theorem

proof of Wagner’s theorem

It is

sufficient

to prove that
the planarity condition given by Wagner’s

theorem

is

equivalent

to the one given by Kuratowski’s theorem,

i.e.

, that a graph

G

=

(

V

,

E

)

has

K

5

or

K

3

,

3

as a minor,
if and only if it has a

subgraph

homeomorphic to either

K

5

or

K

3

,

3

.
It is not restrictive to suppose that

G

is simple and 2-connected.

First, suppose that

G

has a subgraph
homeomorphic to either

K

5

or

K

3

,

3

.
Then there exists

U

⊆

V

such that the subgraph induced by

U

can be transformed into either

K

5

or

K

3

,

3

via a

sequence

of

simple subdivisions

and simple

contractions

through vertices of degree 2.
Since none of these

operations

can alter the number of vertices of degree

d

≠

2

,
and neither

K

5

nor

K

3

,

3

have vertices of degree 2,
none of the simple subdivisions is necessary,
and the homeomorphic subgraph is actually a minor.

For the other

direction

, we prove the following.

1.

If

G

has

K

3

,

3

as a minor,
then

G

has a subgraph homeomorphic to

K

3

,

3

.

2.

If

G

has

K

5

as a minor,
then

G

has a subgraph homeomorphic to either

K

5

or

K

3

,

3

.

Proof of point

1

If

G

has

K

3

,

3

as a minor,
then there exist

U

1

,

U

2

,

U

3

,

W

1

,

W

2

,

W

3

⊆

V

that are

pairwise disjoint

,
induce

connected

subgraphs of

G

,
and such that, for each

i

and

j

,
there exist

u

i

,

j

∈

U

i

and

w

i

,

j

∈

W

j

such that

(

u

i

,

j

,

w

i

,

j

)

∈

E

.
Consequently, for each

i

, there exists a

subtree

of

G

with three leaves,
one leaf in each of the

W

j

’s,
and all of its other nodes inside

U

i

;
the situation with the

j

’s is symmetrical.

As a

consequence

of the

handshake lemma

,
a tree with three leaves is homeomorphic to

K

1

,

3

.
Thus,

G

has a subgraph homeomorphic to six copies of

K

1

,

3

connected three by three,

i.e.

, to

K

3

,

3

.

Proof of point

2

If

G

has

K

5

as a minor,
then there exist pairwise disjoint

U

1

,

…

,

U

5

⊆

V

that induce connected subgraphs of

G

and such that, for every

i

≠

j

,
there exist

u

i

;

{

i

,

j

}

∈

U

i

and

u

j

;

{

i

,

j

}

∈

U

j

such that

(

u

i

;

{

i

,

j

}

,

u

j

;

{

i

,

j

}

)

∈

E

.
Consequently, for each

i

,
there exists a subtree

T

i

of

G

with four leaves,
one leaf in each of the

U

j

’s for

i

≠

j

,
and with all of its other nodes inside

U

i

.

As a consequence of the handshake lemma,
a tree with four leaves is homeomorphic to either

K

1

,

4

or two joint copies of

K

1

,

3

.
If all of the trees above are homeomorphic to

K

1

,

4

,
then

G

has a subgraph homeomorphic to
five copies of

K

1

,

4

, each joint to the others:

i.e.

, to

K

5

.
Otherwise, a subgraph homeomorphic to

K

3

,

3

can be obtained via the following procedure.

1.

Choose one of the

T

i

’s
which is homeomorphic to two joint copies of

K

1

,

3

,
call them

T

i

,

r

and

T

i

,

b

.

2.

Color red the nodes of

T

i

,

r

,
except its two leaves, which are colored blue.

3.

Color blue the nodes of

T

i

,

b

,
except its two leaves, which are colored red.

4.

Color blue the nodes of the

T

j

’s
containing the leaves of

T

i

,

r

.

5.

Color red the nodes of the

T

j

’s
containing the leaves of

T

i

,

b

.

6.

Remove the edges joining nodes with same color in different

T

j

’s.

This “prunes” the

T

j

’s so that they have three leaves,
each in a subgraph of a color different than the rest of their vertices.

The graph formed by the red and blue nodes,
together with the remaining edges,
is then

isomorphic

to

K

3

,

3

.

References

1

Geir Agnarsson, Raymond Greenlaw.

Graph Theory

: Modeling, Applications and

Algorithms

.

Prentice Hall, 2006.

Title

proof of Wagner’s theorem

Canonical name

ProofOfWagnersTheorem

Date of creation

2014-01-16 17:37:41

Last modified on

2014-01-16 17:37:41

Owner

Ziosilvio (18733)

Last modified by

Ziosilvio (18733)

Numerical id

14

Author

Ziosilvio (18733)

Entry type

Proof

Classification

msc 05C99

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