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I would like to now if there has been any work on related problems, that is, shortest path problem in dynamically evolving graphs.

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  • $\begingroup$ As you see, there has been. Can you be more specific in what you are looking for? $\endgroup$
    – András Bátkai
    Commented Jun 20, 2019 at 15:04

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There is a huge literature on shortest paths in dynamic graphs. Try this Google search to start: https://scholar.google.be/scholar?cites=11260993909169863538&as_sdt=2005&sciodt=0,5&hl=en

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