Non-Planar Graphs I know there may be an answer to this somewhere online, if so I haven't found it. But, I was wondering, if there is at least a somewhat intuitive answer for the following: 

Why are all non-planar graphs extensions or super-graphs of $K_{3,3}$ or $K_5$?

I get the proof that $K_{3,3}$ and $K_5$ are nonplanar, but I am still dumbfounded. I feel like with most things in math, there some be some intuition behind theorems, and concepts. But, maybe I am not thinking in the right spot.  
 A: Is there an intuition for the simple finding all the graphs without subdivisions that demand at least one crossing?  
Assume there is such a simple intuition.  Now then, use that to find all the graphs without subdivisions that demand at least two crossings.
Here are two of them, with the first being the Petersen graph, but the seemingly simple intuition is just getting started. It will need to list out the simplest 2-crossing quartic graphs, and other simple graphs that require 2 crossings.  Just for cubic graphs, the assumed-intuition will need to either prove that large cubic 2-crossing graphs will have these two as a subdivision, or that more cubic graphs are needed.
 
Unfortunately -- this minimal set for two crossings is currently an unsolved problem. It's likely computer solvable with a brute-force search through low-order graphs in a few weeks.  Likely there would be a list of 20 or so graphs and a very difficult proof -- in other words, an inelegant result.
No-one can intuitively write down a complete list of all minimal 2-crossing graphs.  Such a level of intuition doesn't exist.
Instead of intuition, suspicion usually works better.  When Kuratowski started looking at this problem, he found $K_{3,3}$ and $K_5$ but he didn't stop there, he was suspicious of it. That helped him to build up the proof. 
Enlightenment usually comes from a process of elimination, and usually there is a lot to eliminate.  It's not using a light switch to examine a large room -- it's crawling blind into the room and feeling every surface in it.  Or programming a computer to do that for you.
Similar -- Snark Theorem: Any Snark has a Petersen Graph minor. These graphs were named snarks by Gardner, Tutte conjectured the theorem. Proven by Robertson, Sanders, Seymour and Thomas in 2001 using methods more difficult than proving the 4-color theorem.
