Are there general tips for determining whether a graph is planar or nonplanar?
If a graph is planar, is there a general approach for finding a planar embedding? I usually try to draw out the vertices one by one, connecting them so that they don't intersect, and if there is an obstruction, I try to find a Kuratowski subgraph. However, I think this method doesn't usually work very well and is very tedious (especially for large graphs).
There are also quite a few theorems on Wikipedia on planarity testing and oftentimes stronger conditions than the basic test using Euler's formula (i.e. that $|E| \leq 3|V|-6$ or $|E|\leq 2|V|-4$ if the girth is greater than $3$) are required. As an example, I was wondering if one could demonstrate the use of a general technique for the following graph, which I think is nonplanar:
Here's what I did. First, I redrew the graph with fewer edge crossings, to make it easier to look for a Hamilton cycle:
I did this by drawing the graph, more or less as pictured, in Geogebra. Then you can drag the points around, and Geogebra adjusts the connecting segments for you. In this form, it's easy to see the Hamilton cycle EGFJDHIBKCAE.
So then I followed the procedure described in the answer I linked in my comment:
and found the graph to be non-planar.