I am trying to show that a graph is planar. Possibly the simplest method I have found is to show the graph can be drawn on a page (i.e. in the plane) without any edges crossing. So, can I assume that if I am given a graph that has edges crossing I can simple move the vertices around to obtain a version such that the edges are not crossing (if such an arrangement exists)?
It may be better to show an example of what I am thinking. Perhaps someone can confirm that what I have done is permutable. Refer to the figure below were I start with the left graph and end with the right graph.
Alternatively, it seems as though one can show a graph is planar if it can be embedded in a disk. I believe the following figure shows such an embedding in a disk