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 2connected.
First, suppose that $G$ has a subgraph homeomorphic to either ${K}_{5}$ or ${K}_{3,3}$. Then there exists $U\subseteq V$ 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 $d\ne 2$, 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 ${U}_{1},{U}_{2},{U}_{3},{W}_{1},{W}_{2},{W}_{3}\subseteq V$ that are pairwise disjoint, induce connected subgraphs of $G$, and such that, for each $i$ and $j$, there exist ${u}_{i,j}\in {U}_{i}$ and ${w}_{i,j}\in {W}_{j}$ such that $({u}_{i,j},{w}_{i,j})\in E$. 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 ${U}_{1},\mathrm{\dots},{U}_{5}\subseteq V$ that induce connected subgraphs of $G$ and such that, for every $i\ne j$, there exist ${u}_{i;\{i,j\}}\in {U}_{i}$ and ${u}_{j;\{i,j\}}\in {U}_{j}$ such that $({u}_{i;\{i,j\}},{u}_{j;\{i,j\}})\in E$. 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 $i\ne j$, 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  20140116 17:37:41 
Last modified on  20140116 17:37:41 
Owner  Ziosilvio (18733) 
Last modified by  Ziosilvio (18733) 
Numerical id  14 
Author  Ziosilvio (18733) 
Entry type  Proof 
Classification  msc 05C99 