Decomposing the search problem into several small problems
The term you are looking for is "divide-and-conquer".  There are gazillions of divide-and-conquer algorithms; any algorithm textbook should show you quite a few examples.  Divide-and-conquer is more general, because it also allows you to do some merger operation to combine the solutions to each piece -- but many divide-and-conquer algorithms do follow the form you are looking for, so I think "divide-and-conquer" should be a helpful search term.

In graphs, one common use of divide-and-conquer is to decompose the graph into connected components or strongly connected components, and then work on each component separately.  In many cases, no pre-processing or post-processing to glue together the solutions is needed: you just take a union of the solution for each component.