[wolfgke]

> how do I identify that a problem is indeed a graph problem?

Mathematical intuition. To answer the next obvious question of how to buld mathematical intuition: Solve lots of math problems (I've also heard that reading "Pólya - How to Solve it" is supposed to be helpful for this; I can't say anything about it).

[marlag]

"Solve lots of math problems"

If you are the type of person that absolutely loved each second of each lecture that your math professor held in high school, where he/she tried to prove an equation on the black board, and you acctually managed to pay attention for long enough to acctually understand what he was talking about, and you got a real kick out of that newly gained intuition, and you now long for that type of "profound" enlightenment, how would yo go about gaining in mathematical intuition when you are in your 40ies?

[Jtsummers]

Mathematical Games, by Martin Gardner.

EDIT: Distractions abound so I hit submit before forgetting.

Why: Martin Gardner wrote on a variety of mathematical topics in fields such as geometry, graph theory, number theory, combinatorics, topology, and beyond. His writing is very approachable, and well sourced. This will help to develop a base vocabulary across the mathematical fields that you can use for further research and investigation of your own once you find the areas that interest you, along with being delightful reads just for their own sake for the mentally curious and engaged.

[wolfgke]

> how would yo go about gaining in mathematical intuition when you are in your 40ies?

Exactly the same way that you would go if you were in your 20ies: Get the relevant textbooks that are typically recommended by professors and read them (sorry, the textbooks that I can recommend for basic studies in mathematics are all in German (my native language); only for main studies in mathematics I can tell English textbooks).