Yes and no...
If you have ultra restricted permissions on DIR-A, and it's owned only by 1 person/group (such as root.root) then only root can get in to see/change DIR-B (in the above example, if root.root owned DIR-A then anyone not having root privileges would not be able to enter DIR-A to ever see DIR-B regardless of DIR-B's permissions).
To access a directory you only need the x bit set, so you could give x writes on DIR-A, they could access DIR-A, and then have the ability to enter/read/write in DIR-B.
HTH
Cool
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