How can I avoid re-computation of Dijkstra algorithm if I add or remove one edge from a graph?
Pathfinding algorithms that are able to recompute the best path when the graph changes without doing a full recomputation are called "Incremental".

The incremental version of Dijkstra's algorithm has the horrible name DynamicSWSF-FP.

The much more common algorithm which combines this with A* is called LPA*