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The D* family of search algorithms are exactly concerned with the updating of shorting paths in dynamically changing graphs. The algorithms were developed for mobile robot path planning problems. Although the algorithms only return the shortest path from the goal to the current robot location, you might be able to use their bookkeeping and updating rules for all-shortest-paths problems too.
You can handle the case where you are deleting an edge/node fairly easily. Just keep track of the actual path between nodes. Then when an edge/node is deleted, go through your paths and see which ones are affected by the change.
Recalculate the shortest paths for these.
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