Postgres - graph iteration - do not iterate twice over nodes - sql

Question

I have a postgres query that traverses a graph. Since it performs some operations one the nodes, I want to track the nodes that have already been visited, to not perform these operations twice.
The documentation shows how to partialy solve the problem by tracking the path in the recursive query (eg: https://www.postgresql.org/docs/9.2/static/queries-with.html). But let's take the following example:
parent_id | child_id
1 | 2
1 | 3
2 | 4
3 | 4
4 | 1
and the example in the documentation:
WITH RECURSIVE search_graph(id, param_1, param_2, param_etc, depth, path, cycle) AS (
SELECT g.id, g.param_1, g.param_2, g.param_etc, 1,
ARRAY[g.id],
false
FROM graph g
UNION ALL
SELECT g.id, g.param_1, g.param_2, g.param_etc, sg.depth + 1,
path || g.id
g.id = ANY(path)
FROM graph g, search_graph sg
WHERE g.id = sg.id AND NOT cycle
)
SELECT * FROM search_graph;
If I understand well, the query shown in the documentation will return two paths: [1, 2, 4, 1] and [1, 3, 4, 1]. Checking for cycles prevents double traversal inside a path (the #1 node will only be seen once) but not across paths: if I query all the nodes, both #1 and #4 will be seen twice (one time in each path).
I would like to know if a query has already been visited independently of the path. I tried to track it by running the recursive query in a function to which an array "seen" would be passed as argument. The idea was to append to this array and not reinitialize the path every time.
I tried this, without success
CREATE OR REPLACE FUNCTION recursive_function(id int, seen int[] DEFAULT '{}'::int[])
RETURNS TABLE (
id int,
param_1 text,
...
) AS
$func$
BEGIN
RETURN QUERY
WITH RECURSIVE search_graph(id, param_1, param_2, param_etc, path, cycle) AS (
SELECT g.id, g.param_1, g.param_2, g.param_etc,
-- here I try to update a variable that would be common to the loop
-- but it does not work
seen || ARRAY[g.id],
false,
FROM computations_i_do_not_want_to_run_twice() g
UNION ALL
SELECT g.id, g.param_1, g.param_2, g.param_etc,
seen || g.id
g.id = ANY(seen)
FROM computations_i_do_not_want_to_run_twice() q, search_graph sg
WHERE g.id = sg.id AND NOT cycle
)
SELECT * FROM search_graph;
END
$func$ LANGUAGE plpgsql;
SELECT
*
FROM
recursive_function(123);
Note that I cannot iterate the graph backwards, starting with the children. Any idea?