MySQL: How to determine foreign key relationships programmatically?

There are two tables you can query to get this information: INFORMATION_SCHEMA.TABLE_CONSTRAINTS and INFORMATION_SCHEMA.KEY_COLUMN_USAGE.

Here's a query from the comments on the latter page linked above, which demonstrates how to get the info you seek.

SELECT CONCAT( table_name, '.', column_name, ' -> ', 
  referenced_table_name, '.', referenced_column_name ) AS list_of_fks 
FROM INFORMATION_SCHEMA.key_column_usage 
WHERE referenced_table_schema = 'test' 
  AND referenced_table_name IS NOT NULL 
ORDER BY table_name, column_name;

Use your schema name instead of 'test' above.

Here you have a little improvement over the @bill solution:

SELECT CONSTRAINT_SCHEMA AS db,
       CONCAT (
           TABLE_NAME,
           '.',
           COLUMN_NAME,
           ' -> ',
           REFERENCED_TABLE_NAME,
           '.',
           REFERENCED_COLUMN_NAME
       ) AS relationship 
FROM INFORMATION_SCHEMA.KEY_COLUMN_USAGE
WHERE REFERENCED_TABLE_NAME = 'your_table_name'
ORDER BY CONSTRAINT_SCHEMA,
         TABLE_NAME,
         COLUMN_NAME;

In this case I was filtering by relationships with the "your_table_name" fields and seeing from which database the relationship comes.