How to efficiently find linked items by path in MongoDB tree using $graphLookup?

Question

I would like to find documents on a path in a tree structure in MongoDB using $graphLookup.

For example, let's say I have this small structure:

0 - root
└── 01 "home"
     ├── 011 "john"
     │    ├── 0111 "documents"
     │    └── 0112 "movies"
     │         ├── 01121 "Holiday inn"
     │         └── 01122 "Harry Potter"
     └── 012 "maria"

representing the data:

[
  {"_id":"0","name":"root","parent":""},
  {"_id":"01","name":"home","parent":"0"},
  {"_id":"011","name":"john","parent":"01"},
  {"_id":"0111","name":"documents","parent":"011"},
  {"_id":"0112","name":"movies","parent":"011"},
  {"_id":"01121","name":"Holiday Inn","parent":"0112"},
  {"_id":"01122","name":"Harry Potter","parent":"0112"},
  {"_id":"012","name":"maria","parent":"01"}
]

Is it possible to use $graphLookup (potentially with other steps in the aggregation) to

• check that a given path (e.g. "root/home/maria") exists?
• to retrieve all documents on a target path (e.g. "root/home/john/movies/Harry Potter")?

In principle it should be possible to do it. But I'm not sure if $graphLookup is able to do it efficiently. That is, by checking that on each step in the recursion we're looking for a document with a specific name. The inefficient alternative would be to generate all paths of length N (starting with "root" folder) and find the matching one. Another inefficient options would be to store the path in the documents themselves, but that would incur a cost of many updates to child documents in case of renames. These are solutions I would very much like to avoid, for obvious reasons. But I'm unable to figure out whether an efficient query can be written for MongoDB.

Complications:

• a folder may have potentially many children, so it may be easier to model the structure using parent references,
• users may rename things, so it wouldn't be efficient to store the path in the documents themselves.