Recently when I read the proof of the Kuratowski’s Theorem, I was stuck at the final step where it States 4 cases (Since this proof uses many lemmas, it is difficult to give one context related with lemmas in the question box).
I read this lecture_1 p7 which is same as this video 12:10, but they both don't show why there are only these 4 cases
We claim, without proof (although you should be able to prove this!) that the impedimentary structure inside C must take one of the forms illustrated in Figure 8
This paper p10,11 seems to drop the top-right case in the lecture_1.
It is trivial to check these 4 cases are impedimentary structures which avoids $uv$ to be outside so that $G$ can be planar which is contradiction
Now, it must be the case that we could not have placed the path P interior to C, otherwise we would do so and thus be able to include the edge uv exterior to C. Hence, there must be some impedimentary structure inside C blocking us from drawing P in the interior.
Q:
How to prove that there is only the 4 cases listed in lecture_1 possible to be the impedimentary structure?