Get all the rows referencing (via foreign keys) a particular row in a table

You can do one of the following:

1) Add reference_count field to master table. Using triggers on detail tables increase the reference count whenever a row with this master_id is added. Decrease the count, when row gets deleted. When reference_count reaches 0 - delete the record.

2) Use pg_constraint table (details here) to get the list of referencing tables and create a dynamic SQL query.

3) Create triggers on every detail table, that deletes master_id in main table. Silence error messages with BEGIN ... EXCEPTION ... END.

In case someone wants a real count of rows in all other tables that reference a given master row, here is some PL/pgSQL. Note that this works in plain case with single column constraints. It gets more involved for multi-column constraints.

CREATE OR REPLACE FUNCTION count_references(master regclass, pkey_value integer,
    OUT "table" regclass, OUT count integer)
    RETURNS SETOF record 
    LANGUAGE 'plpgsql'
    VOLATILE 
AS $BODY$
declare
  x record;           -- constraint info for each table in question that references master
  sql text;           -- temporary buffer
begin
  for x in
    select conrelid, attname
    from pg_constraint
    join pg_attribute on conrelid=attrelid and attnum=conkey[1]
    where contype='f' and confrelid=master
      and confkey=( -- here we assume that FK references master's PK
        select conkey
        from pg_constraint
        where conrelid=master and contype='p'
      )
  loop
    "table" = x.conrelid;
    sql = format('select count(*) from only %s where %I=$1', "table", x.attname);
    execute sql into "count" using pkey_value;
    return next;
  end loop;
end
$BODY$;

Then use it like

select * from count_references('master', 1) where count>0

This will return a list of tables that have references to master table with id=1.