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I'm interested in creating a mathematical model of an underground (tube) system. The overall aim is being able to efficiently calculate travel times between stations.

Such a system has some interesting properties, for example:

  • if station $B$ belongs to the quickest route between $A$ and $C$, then the sub-routes $AB$ and $BC$ are also the quickest;
  • however, in general, the opposite is not true. If $AB$ and $BC$ are quickest routes, then it does not follow that the union is also the quickest route.

What is the branch of mathematics that studies such problems? What are some interesting theorems?

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    Watford Junction is on the quickest route for a train journey from London Euston to Glasgow. However, that;s not the quickest route from London Euston to Watford Junction, because the Glasgow trains only allow passengers to board at Watford, and you can't get off.2011-03-30
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    @MikeScott in most graph descriptions of such a situation, there would be two (or more) distinct nodes for Watford Junction for each such behavior.2013-03-18

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Graph theory and Combinatorial optimization. See e.g. http://en.wikipedia.org/wiki/Shortest_path_problem