Textbook or video lectures to develope intuition for solving any unseen Graph Theory problems
I'm writing only a partial answer, because I currently lack time. That means that I'm not properly sourcing everything, and largely working from my own memory.  I don't believe that anything I say below will be wrong, though it is definitely possible that I might be forgetting a few things.  In any case, hopefully this will be enough to get you started / unstuck.
The test does look quite difficult!  I teach several of the topics, and they were largely what I would expect from the end of an undergraduate university degree, but with the added pressure of timing and high stakes.  Ouch!
When attempting to gain fluency (in any topic), you are looking to make a few kinds of connections:

&quot;This is just like...&quot; -- these connections allow you to utilize systems that your brain already understands well.

&quot;These ideas group together&quot; -- this is called &quot;chunking&quot;, and it is how we group small ideas into larger and larger pictures.  When you &quot;chunk&quot; ideas together, you are able to manipulate the chunk in your mind instead of all of the small details.

&quot;This system looks like&quot; -- much like #1, this allows you to &quot;see&quot; how systems work in your mind's eye.  Then you can manipulate your visuals.


In essence, you are trying to group and simplify ideas.
To gain fluency, additionally, don't be afraid of memorization (though don't focus on it exclusively, either.)  The purpose of memorization should not be to be able to regurgitate everything, but rather to give you quick enough recall that you can use the idea in a larger system.
Think about it this way: there is a cognitive tax to using any idea that we know.  If the cognitive tax is very high, then using the idea takes nearly all of our focus.  This limits what we can learn, because to really understand larger ideas, we need space in the active, working parts of our brain to encompass all of the smaller ideas that the larger idea needs.  If the smaller ideas are taking up all of our active resources, that becomes impossible.
Thus, the goal of memorization is to reduce cognitive tax.  If you are working hard to memorize beyond that, then you are wasting your efforts - you are not going to gain further fluency through memorization.  You need to instead focus on chunking, visualizing, and relating ideas to one another so that you can see bigger pictures and larger groups.
Also, it is useful to practice on a mix of simple and hard problems.  The first group reinforce and strengthen the patterned connections that you have already made.  The second group force you to stretch those patterns out in new directions, and can help you spot new uses and simplifications.
I have always found that teaching others helps me to gain fluency, as it forces me to make explanations that distill systems for other people.
Good luck!