Postgres 12

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-- Test setup --
CREATE TABLE master (master_id int PRIMARY KEY);

INSERT INTO master VALUES (1), (2);

CREATE TABLE other (
   other_id int PRIMARY KEY,
   master_id int NOT NULL REFERENCES master (master_id) ON DELETE RESTRICT    
);

INSERT INTO other VALUES (1,1);      -- only master_id 1 is referenced

-- Function --
CREATE OR REPLACE FUNCTION f_can_del(_id int)
  RETURNS boolean
  LANGUAGE plpgsql AS 
$func$
BEGIN
   DELETE FROM master WHERE master_id = _id;  -- Test-DELETE (will be rolled back)

   IF NOT FOUND THEN
      RETURN NULL;                            -- ID not found, return NULL
   END IF;

   RAISE SQLSTATE 'MYERR';                    -- If DELETE, raise custom exception

   EXCEPTION
      WHEN FOREIGN_KEY_VIOLATION THEN
         RETURN FALSE;

      WHEN SQLSTATE 'MYERR' THEN
         RETURN TRUE;
      -- other exceptions are propagated as usual
END  
$func$;

CREATE TABLE

INSERT 0 2

CREATE TABLE

INSERT 0 1

CREATE FUNCTION

SELECT f_can_del(1)  AS test1
     , f_can_del(2)  AS test2
     , f_can_del(23) AS test23;

test1
test2
test23

f
t
null

SELECT 1

TABLE master;

master_id

1

2

SELECT 2