Graph Theory 


with labelled vertices 
It all started with Leonhard Euler and the Bridges of Koenigsberg problem. In that problem the only thing that mattered was the pieces of land that were connected. The exact shape, placing and lengths of the bridges were irrelevant.
Graph Theory is also the area with one of the most common examples of Pure Maths found in school, namely, the Four Colour Theorem.
so that regions sharing a border get different colours. How many colours do you need? 
This question was first posed in 1852, and it wasn't until 1976 that a proof was finally given. Even then there was, and still is, some controversy, because the proof requires a computer to check a large number of subcases. These then have to be combined in a clever way  the computer doesn't actually do the proof  but even so, it's not a proof in the traditional sense.
You can read more about the Four Colour Theorem and Graph Theory in general here:
Another problem that turns out to be an example in graph theory is on the Dominoes Unlimited page.
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Quotation from Tim BernersLee 