Hey Bioho4,
It is a very time consuming question.
Usually when verifying if something is planar, I use the euler's formula, v-e+f=2.
However the problem with this shortcut is that to tell the number of faces, it needs to be re-drawn with no edges overlapping, which obviously makes the shortcut a mute point.
In practise exams, I never really draw them out. I visualise in my head how to make it planar. This is much easier for the last two in this question, but it is just going through combinations of how you could re-arrange the edges to not overlap.
While it is difficult in your head or on paper, I have found it gets much easier with time and practise. Usually exam 1's will not be very time-stressful anyway, and I haven't come across any exam 2 type questions of these sorts.
Maybe someone else has a much better method, but I haven't come across it in over 30+ further exams so far.
If they do, I'll be keen to know it.
