Is processing required dimensions enough or to process the entire cube?
I think you are confusing things.

First of all, processing the cube and processing the dimensions is sort of separate. I think you mean "processing the entire database".

If you process only dimensions, and not the cube, the measures will never be updated.

You could Process Update the dimensions which wouldn't unprocess the cube but would drop the indexes and flexible aggregations so you would need to Process Index in the same transaction (to avoid the indexes being created on data that may have changed since the process dimension).

Also if you fullprocess dimensions, the cubes where the dimensions are linked will be in an "unprocessed" state.

On the other hand. You don't necessarily need to process the entire database, If you are absolutely sure nothing in one of the underlying dimension tables changes you can get away with just processing some of the dimensions and the cube without processing the entire database.

On the other hand, what makes you ask this question? It's certainly not something I would recommend as there are probably better solutions, such as processing incremental which won't do a lot if nothing has changed.