Is recursive query an option for concatenating text for building a path

Simplifying a bit. I have the following data structure:

| id       | name           | traversal_ids[] |
| -------- | -------------- |-----------------|
| 1        | container1     | {1}             |
| 2        | container2     | {2}             |
| 3        | subcontainer1  | {2, 3}          |
| 4        | container4     | {4}             |
| 5        | subsub         | {2 ,3 ,5}       |

and I'd like to transform this in the following:

| id       | name           | path                             |
|--------  |--------------  |----------------------------------|
| 1        | container1     | /container1                      |
| 2        | container2     | /container2                      |
| 3        | subcontainer1  | /container2/subcontainer1        |
| 4        | container4     | /container4                      |
| 5        | subsub         | /container2/subcontainer1/subsub |   

I'd like as much as possible to give this to the database to execute, but unfortunately I lack the sql skills. If it's not possible I'll try to get this done inside the app.

Can this be achieved in SQL?

Thanks!

No need for recursion. We can unnest the array in a lateral join, bring the corresponding names with a join, then aggregate back:

select t.*, x.*
from mytable t
cross join lateral (
    select string_agg(t1.name, '/' order by x.ord) path
    from unnest(t. traversal_ids) with ordinality x(id, ord)
    inner join mytable t1 on t1.id = x.id
) x

Assuming the traversal ids are always with ascendant order, You can do it using a simple inner join with ANY operator as follows :

with cte as (
  select t2.id, t2.name, t1.name as sub_name, t1.id as sub_id
  from mytable t1
  inner join mytable t2 on t1.id = ANY(t2.traversal_ids)
)
select id, name, '/' || string_agg(sub_name, '/' order by sub_id) as path
from cte
group by id, name 

Demo here